Razlika između verzija stranice "Parabola (matematika)"

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class="vector-toc-numb">1.1.1.1</span> <span>Kanonski oblik jednačine</span> </div> </a> <ul id="toc-Kanonski_oblik_jednačine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jednačina_konusnog_presjeka" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Jednačina_konusnog_presjeka"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1.2</span> <span>Jednačina konusnog presjeka</span> </div> </a> <ul id="toc-Jednačina_konusnog_presjeka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Karakteristike_parabole_u_odnosu_na_njen_položaj" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Karakteristike_parabole_u_odnosu_na_njen_položaj"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1.3</span> <span>Karakteristike parabole u odnosu na njen položaj</span> </div> </a> <ul id="toc-Karakteristike_parabole_u_odnosu_na_njen_položaj-sublist" class="vector-toc-list"> </ul> </li> <li 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class="vector-toc-numb">2</span> <span>Parabola u relnom svijetu</span> </div> </a> <ul id="toc-Parabola_u_relnom_svijetu-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Također_pogledajte" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Također_pogledajte"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Također pogledajte</span> </div> </a> <ul id="toc-Također_pogledajte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vanjski_linkovi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vanjski_linkovi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vanjski linkovi</span> </div> </a> <ul id="toc-Vanjski_linkovi-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Uključi/isključi sadržaj" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Uključi/isključi sadržaj</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Razlika između verzija stranice "Parabola (matematika)"</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Idi na članak na drugom jeziku. Dostupno na 87 jezika" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-87" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">87 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Parabool" title="Parabool – afrikans" lang="af" hreflang="af" data-title="Parabool" data-language-autonym="Afrikaans" data-language-local-name="afrikans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%A3%E1%88%8B" title="ባላ – amharski" lang="am" hreflang="am" data-title="ባላ" data-language-autonym="አማርኛ" data-language-local-name="amharski" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%B7%D8%B9_%D9%85%D9%83%D8%A7%D9%81%D8%A6" title="قطع مكافئ – arapski" lang="ar" hreflang="ar" data-title="قطع مكافئ" data-language-autonym="العربية" data-language-local-name="arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Parabola" title="Parabola – azerbejdžanski" lang="az" hreflang="az" data-title="Parabola" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – baškirski" lang="ba" hreflang="ba" data-title="Парабола" data-language-autonym="Башҡортса" data-language-local-name="baškirski" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%B0%D0%BB%D0%B0" title="Парабала – bjeloruski" lang="be" hreflang="be" data-title="Парабала" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%B0%D0%BB%D0%B0" title="Парабала – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Парабала" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – bugarski" lang="bg" hreflang="bg" data-title="Парабола" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%B0%E0%A6%BE%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="পরাবৃত্ত – bengalski" lang="bn" hreflang="bn" data-title="পরাবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="bengalski" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Par%C3%A0bola" title="Paràbola – katalonski" lang="ca" hreflang="ca" data-title="Paràbola" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/P%C4%83u-%C5%ADk-si%C3%A1ng" title="Pău-ŭk-siáng – Mindong" lang="cdo" hreflang="cdo" data-title="Pău-ŭk-siáng" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%DB%95%D9%88%D8%A7%D9%86%DB%95%D8%A8%DA%95" title="کەوانەبڕ – centralnokurdski" lang="ckb" hreflang="ckb" data-title="کەوانەبڕ" data-language-autonym="کوردی" data-language-local-name="centralnokurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Parabola_(matematika)" title="Parabola (matematika) – češki" lang="cs" hreflang="cs" data-title="Parabola (matematika)" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – čuvaški" lang="cv" hreflang="cv" data-title="Парабола" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Parabola" title="Parabola – velški" lang="cy" hreflang="cy" data-title="Parabola" data-language-autonym="Cymraeg" data-language-local-name="velški" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Parabel" title="Parabel – danski" lang="da" hreflang="da" data-title="Parabel" data-language-autonym="Dansk" data-language-local-name="danski" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Parabel_(Mathematik)" title="Parabel (Mathematik) – njemački" lang="de" hreflang="de" data-title="Parabel (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Parabol" title="Parabol – Zazaki" lang="diq" hreflang="diq" data-title="Parabol" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CE%B2%CE%BF%CE%BB%CE%AE_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Παραβολή (γεωμετρία) – grčki" lang="el" hreflang="el" data-title="Παραβολή (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Parabola" title="Parabola – engleski" lang="en" hreflang="en" data-title="Parabola" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Parabolo_(matematiko)" title="Parabolo (matematiko) – esperanto" lang="eo" hreflang="eo" data-title="Parabolo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Par%C3%A1bola_(matem%C3%A1tica)" title="Parábola (matemática) – španski" lang="es" hreflang="es" data-title="Parábola (matemática)" data-language-autonym="Español" data-language-local-name="španski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Parabool" title="Parabool – estonski" lang="et" hreflang="et" data-title="Parabool" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Parabola_(matematika)" title="Parabola (matematika) – baskijski" lang="eu" hreflang="eu" data-title="Parabola (matematika)" data-language-autonym="Euskara" data-language-local-name="baskijski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B3%D9%87%D9%85%DB%8C" title="سهمی – perzijski" lang="fa" hreflang="fa" data-title="سهمی" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Paraabeli" title="Paraabeli – finski" lang="fi" hreflang="fi" data-title="Paraabeli" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Parabil" title="Parabil – farski" lang="fo" hreflang="fo" data-title="Parabil" data-language-autonym="Føroyskt" data-language-local-name="farski" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Parabole" title="Parabole – francuski" lang="fr" hreflang="fr" data-title="Parabole" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Paraabel" title="Paraabel – sjeverni frizijski" lang="frr" hreflang="frr" data-title="Paraabel" data-language-autonym="Nordfriisk" data-language-local-name="sjeverni frizijski" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Parab%C3%B3il" title="Parabóil – irski" lang="ga" hreflang="ga" data-title="Parabóil" data-language-autonym="Gaeilge" data-language-local-name="irski" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Parab%C3%B2la" title="Parabòla – škotski galski" lang="gd" hreflang="gd" data-title="Parabòla" data-language-autonym="Gàidhlig" data-language-local-name="škotski galski" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Par%C3%A1bola_(xeometr%C3%ADa)" title="Parábola (xeometría) – galicijski" lang="gl" hreflang="gl" data-title="Parábola (xeometría)" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Palapola" title="Palapola – havajski" lang="haw" hreflang="haw" data-title="Palapola" data-language-autonym="Hawaiʻi" data-language-local-name="havajski" class="interlanguage-link-target"><span>Hawaiʻi</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A8%D7%91%D7%95%D7%9C%D7%94" title="פרבולה – hebrejski" lang="he" hreflang="he" data-title="פרבולה" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%B5%E0%A4%B2%E0%A4%AF" title="परवलय – hindi" lang="hi" hreflang="hi" data-title="परवलय" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Parabola" title="Parabola – hrvatski" lang="hr" hreflang="hr" data-title="Parabola" data-language-autonym="Hrvatski" data-language-local-name="hrvatski" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Parabola_(g%C3%B6rbe)" title="Parabola (görbe) – mađarski" lang="hu" hreflang="hu" data-title="Parabola (görbe)" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%A1%D6%80%D5%A1%D5%A2%D5%B8%D5%AC" title="Պարաբոլ – armenski" lang="hy" hreflang="hy" data-title="Պարաբոլ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Parabola" title="Parabola – indonezijski" lang="id" hreflang="id" data-title="Parabola" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Parabolo" title="Parabolo – ido" lang="io" hreflang="io" data-title="Parabolo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fleygbogi" title="Fleygbogi – islandski" lang="is" hreflang="is" data-title="Fleygbogi" data-language-autonym="Íslenska" data-language-local-name="islandski" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Parabola_(geometria)" title="Parabola (geometria) – italijanski" lang="it" hreflang="it" data-title="Parabola (geometria)" data-language-autonym="Italiano" data-language-local-name="italijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%94%BE%E7%89%A9%E7%B7%9A" title="放物線 – japanski" lang="ja" hreflang="ja" data-title="放物線" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%90%E1%83%A0%E1%83%90%E1%83%91%E1%83%9D%E1%83%9A%E1%83%90" title="პარაბოლა – gruzijski" lang="ka" hreflang="ka" data-title="პარაბოლა" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – kazaški" lang="kk" hreflang="kk" data-title="Парабола" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%94%E1%9F%89%E1%9E%B6%E1%9E%9A%E1%9F%89%E1%9E%B6%E1%9E%94%E1%9E%BC%E1%9E%9B" title="ប៉ារ៉ាបូល – kmerski" lang="km" hreflang="km" data-title="ប៉ារ៉ាបូល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kmerski" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%AC%EB%AC%BC%EC%84%A0" title="포물선 – korejski" lang="ko" hreflang="ko" data-title="포물선" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – kirgiški" lang="ky" hreflang="ky" data-title="Парабола" data-language-autonym="Кыргызча" data-language-local-name="kirgiški" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Parabola" title="Parabola – latinski" lang="la" hreflang="la" data-title="Parabola" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Parabol%C4%97" title="Parabolė – litvanski" lang="lt" hreflang="lt" data-title="Parabolė" data-language-autonym="Lietuvių" data-language-local-name="litvanski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Parabola" title="Parabola – latvijski" lang="lv" hreflang="lv" data-title="Parabola" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – makedonski" lang="mk" hreflang="mk" data-title="Парабола" data-language-autonym="Македонски" data-language-local-name="makedonski" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B4%B0%E0%B4%BE%E0%B4%AC%E0%B5%8A%E0%B4%B3_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="പരാബൊള (ഗണിതം) – malajalam" lang="ml" hreflang="ml" data-title="പരാബൊള (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="malajalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%B5%E0%A4%B2%E0%A4%AF" title="परवलय – marati" lang="mr" hreflang="mr" data-title="परवलय" data-language-autonym="मराठी" data-language-local-name="marati" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Parabola" title="Parabola – malajski" lang="ms" hreflang="ms" data-title="Parabola" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Parabool_(wiskunde)" title="Parabool (wiskunde) – nizozemski" lang="nl" hreflang="nl" data-title="Parabool (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Parabel" title="Parabel – norveški (Nynorsk)" lang="nn" hreflang="nn" data-title="Parabel" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Parabel" title="Parabel – norveški (Bokmal)" lang="nb" hreflang="nb" data-title="Parabel" data-language-autonym="Norsk bokmål" data-language-local-name="norveški (Bokmal)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Parab%C3%B2la" title="Parabòla – oksitanski" lang="oc" hreflang="oc" data-title="Parabòla" data-language-autonym="Occitan" data-language-local-name="oksitanski" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Baallaa" title="Baallaa – oromo" lang="om" hreflang="om" data-title="Baallaa" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Parabola_(matematyka)" title="Parabola (matematyka) – poljski" lang="pl" hreflang="pl" data-title="Parabola (matematyka)" data-language-autonym="Polski" data-language-local-name="poljski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Par%C3%A0bola" title="Paràbola – Piedmontese" lang="pms" hreflang="pms" data-title="Paràbola" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Par%C3%A1bola" title="Parábola – portugalski" lang="pt" hreflang="pt" data-title="Parábola" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Parabol%C4%83" title="Parabolă – rumunski" lang="ro" hreflang="ro" data-title="Parabolă" data-language-autonym="Română" data-language-local-name="rumunski" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – ruski" lang="ru" hreflang="ru" data-title="Парабола" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – Rusyn" lang="rue" hreflang="rue" data-title="Парабола" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%B5%E0%A4%B2%E0%A4%AF%E0%A4%83" title="परवलयः – sanskrit" lang="sa" hreflang="sa" data-title="परवलयः" data-language-autonym="संस्कृतम्" data-language-local-name="sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Par%C3%A0bbula_(matim%C3%A0tica)" title="Paràbbula (matimàtica) – sicilijanski" lang="scn" hreflang="scn" data-title="Paràbbula (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="sicilijanski" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Parabola" title="Parabola – srpskohrvatski" lang="sh" hreflang="sh" data-title="Parabola" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpskohrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Parabola" title="Parabola – Simple English" lang="en-simple" hreflang="en-simple" data-title="Parabola" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Parabola" title="Parabola – slovački" lang="sk" hreflang="sk" data-title="Parabola" data-language-autonym="Slovenčina" data-language-local-name="slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Parabola" title="Parabola – slovenski" lang="sl" hreflang="sl" data-title="Parabola" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Parabola" title="Parabola – albanski" lang="sq" hreflang="sq" data-title="Parabola" data-language-autonym="Shqip" data-language-local-name="albanski" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – srpski" lang="sr" hreflang="sr" data-title="Парабола" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Parabel_(kurva)" title="Parabel (kurva) – švedski" lang="sv" hreflang="sv" data-title="Parabel (kurva)" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%B5%E0%AF%81" title="பரவளைவு – tamilski" lang="ta" hreflang="ta" data-title="பரவளைவு" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%B2%E0%B8%A3%E0%B8%B2%E0%B9%82%E0%B8%9A%E0%B8%A5%E0%B8%B2" title="พาราโบลา – tajlandski" lang="th" hreflang="th" data-title="พาราโบลา" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Parabol" title="Parabol – turski" lang="tr" hreflang="tr" data-title="Parabol" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B1%D0%BE%D0%BB%D0%B0" title="Парабола – ukrajinski" lang="uk" hreflang="uk" data-title="Парабола" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Parabola_(chiziq)" title="Parabola (chiziq) – uzbečki" lang="uz" hreflang="uz" data-title="Parabola (chiziq)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Parabol" title="Parabol – vijetnamski" lang="vi" hreflang="vi" data-title="Parabol" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Parabola_(matematika)" title="Parabola (matematika) – varej" lang="war" hreflang="war" data-title="Parabola (matematika)" data-language-autonym="Winaray" data-language-local-name="varej" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%8A%9B%E7%89%A9%E7%BA%BF" title="抛物线 – Wu kineski" lang="wuu" hreflang="wuu" data-title="抛物线" data-language-autonym="吴语" data-language-local-name="Wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%90%D7%A8%D7%90%D7%91%D7%A2%D7%9C" title="פאראבעל – jidiš" lang="yi" hreflang="yi" data-title="פאראבעל" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8A%9B%E7%89%A9%E7%BA%BF" title="抛物线 – kineski" lang="zh" hreflang="zh" data-title="抛物线" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%8B%8B%E7%89%A9%E7%B7%9A" title="拋物線 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="拋物線" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%8B%8B%E7%89%A9%E7%B6%AB" title="拋物綫 – kantonski" lang="yue" hreflang="yue" data-title="拋物綫" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q48297#sitelinks-wikipedia" title="Uredi međujezičke linkove" class="wbc-editpage">Uredi veze</a></span></div> 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title="Wikipedia:Administratori">Administratori</a></div><div class="mw-diff-usereditcount"><span>385.142</span> edits</div></div></div><div id="mw-diff-otitle3"><abbr class="minoredit" title="Manja izmjena">m</abbr> <span class="comment comment--without-parentheses">Lokalizacija (left,right,center,thumbnail,Image)</span></div><div id="mw-diff-otitle5"></div><div id="mw-diff-otitle4"><a href="/w/index.php?title=Parabola_(matematika)&diff=prev&oldid=1716598" title="Parabola (matematika)" id="differences-prevlink">← Starija izmjena</a></div></td> <td colspan="2" class="diff-ntitle diff-side-added"><div id="mw-diff-ntitle1"><strong><a href="/w/index.php?title=Parabola_(matematika)&oldid=1722722" title="Parabola (matematika)">Verzija na dan 10 mart 2012 u 00:00</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Parabola_(matematika)&action=edit&oldid=1722722" title="Parabola (matematika)">uredi</a></span><span class="mw-diff-timestamp" data-timestamp="2012-03-09T23:00:14Z"></span> <span class="mw-diff-undo"><a href="/w/index.php?title=Parabola_(matematika)&action=edit&undoafter=1716598&undo=1722722" title="Dugme "poništi" poništava ovu izmjenu, te otvara stranicu u režimu za uređivanje s pregledom izmjena koje će biti poništene. Također omogućava unošenje razloga poništenja u sažetak.">poništi</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/wiki/Korisnik:EdinBot" class="mw-userlink" title="Korisnik:EdinBot" data-mw-revid="1722722"><bdi>EdinBot</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/Razgovor_s_korisnikom:EdinBot" class="mw-redirect mw-usertoollinks-talk" title="Razgovor s korisnikom:EdinBot">razgovor</a> | <a href="/wiki/Posebno:Doprinosi/EdinBot" class="mw-usertoollinks-contribs" title="Posebno:Doprinosi/EdinBot">doprinosi</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"><a href="/wiki/Wikipedia:Botovi" class="mw-redirect" title="Wikipedia:Botovi">Botovi</a>, <a href="/wiki/Wikipedia:Urednik" class="mw-redirect" title="Wikipedia:Urednik">Urednici</a>, <a href="/wiki/Wikipedia:Administratori" title="Wikipedia:Administratori">Administratori</a></div><div class="mw-diff-usereditcount"><span>385.142</span> edits</div></div></div><div id="mw-diff-ntitle3"><abbr class="minoredit" title="Manja izmjena">m</abbr> <span class="comment comment--without-parentheses">Bot: Standardizacija poglavlja</span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"><a href="/w/index.php?title=Parabola_(matematika)&diff=next&oldid=1722722" title="Parabola (matematika)" id="differences-nextlink">Novija izmjena →</a></div></td> </tr><tr> <td colspan="2" class="diff-lineno">Red 144:</td> <td colspan="2" class="diff-lineno">Red 144:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>Ako se [[zraka]] koja prilazi paraboli (ili [[paraboloid]]u) paralelno sa osom simetrije odbije od parabole/paraboloida, prolazit će fokusom. To je razlog, zašto se proizvode [[paraboličko ogledalo|parabolička ogledala]] i [[antena|antene]] (npr. kod [[automobil]]a, [[dvogled]]a, [[satelit|telekomunikacijskih satelita]] i sl.).</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>Ako se [[zraka]] koja prilazi paraboli (ili [[paraboloid]]u) paralelno sa osom simetrije odbije od parabole/paraboloida, prolazit će fokusom. To je razlog, zašto se proizvode [[paraboličko ogledalo|parabolička ogledala]] i [[antena|antene]] (npr. kod [[automobil]]a, [[dvogled]]a, [[satelit|telekomunikacijskih satelita]] i sl.).</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>== <del class="diffchange diffchange-inline">Relevantni</del> <del class="diffchange diffchange-inline">članci</del> ==</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== <ins class="diffchange diffchange-inline">Također</ins> <ins class="diffchange diffchange-inline">pogledajte</ins> ==</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* [[Kružnica]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* [[Kružnica]]</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>* [[Hiperbola]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>* [[Hiperbola]]</div></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Verzija na dan 10 mart 2012 u 00:00</h2> <div class="mw-content-ltr mw-parser-output" lang="bs" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Parabola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Parabola.svg/220px-Parabola.svg.png" decoding="async" width="220" height="266" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Parabola.svg/330px-Parabola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Parabola.svg/440px-Parabola.svg.png 2x" data-file-width="400" data-file-height="484" /></a><figcaption>Parabola</figcaption></figure> <p><b>Parabola</b> je vrsta <a href="/wiki/Konusni_presjek" title="Konusni presjek">konusnog presjeka</a>, <a href="/wiki/Kriva" title="Kriva">kriva</a> drugog stepena. Parabola je <a href="/wiki/Skup" class="mw-redirect" title="Skup">skup</a> takvih <a href="/w/index.php?title=To%C4%8Dka&action=edit&redlink=1" class="new" title="Točka (stranica ne postoji)">tačaka</a> u ravni, koje su jednako udaljene od <a href="/w/index.php?title=Prava&action=edit&redlink=1" class="new" title="Prava (stranica ne postoji)">prave</a> (tzv. <i>direktrisa</i>) od date tačke, koja leži na toj pravoj (tzv. <i>fokus</i>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Osobine">Osobine</h2></div> <p>Parabola je osno <a href="/wiki/Simetrija" title="Simetrija">simetrična</a>. Osa simetrije prolazi fokusom parabole i okomita je na direktrisu. Rotacijom parabole oko njene ose simetrije nastaje <a href="/wiki/Paraboloid" title="Paraboloid">paraboloid</a>. </p><p>Za parabolu kažemo , da je u normalnom položaju, kada je njena osa paralelna s osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ili <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. </p><p>Parabola se može definisati kao konusni presjek s nagibom koji je jednak jedan. Iz tog proizilazi, da su sve parabole <a href="/w/index.php?title=Sli%C4%8Dnost&action=edit&redlink=1" class="new" title="Sličnost (stranica ne postoji)">slične</a>. Parabolu možemo shvatiti kao granicu <a href="/w/index.php?title=Niz_(matematika)&action=edit&redlink=1" class="new" title="Niz (matematika) (stranica ne postoji)">niza</a> <a href="/wiki/Elipsa" title="Elipsa">elipse</a>, u kojoj je jednan od fokusa stacionaran, a drugi se postepeno udaljava do beskonačnosti. </p> <div class="mw-heading mw-heading3"><h3 id="Matematički_zapisi"><span id="Matemati.C4.8Dki_zapisi"></span>Matematički zapisi</h3></div> <p><b>Implicitni zapis</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|XF\|=\|Xd\|\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mi>F</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mi>d</mi> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|XF\|=\|Xd\|\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a49dae853ed0177867b17307177acdd96aef4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:15.052ex; height:2.843ex;" alt="{\displaystyle \|XF\|=\|Xd\|\,\!}"></span></dd></dl> <p><a href="/wiki/Skup" class="mw-redirect" title="Skup">Skup</a> svih <a href="/wiki/Ta%C4%8Dka" class="mw-disambig" title="Tačka">tačaka</a> <i>X</i> u <a href="/wiki/Ravan" title="Ravan">ravni</a>, koje imaju istu udaljenost od <a href="/wiki/Fokus" title="Fokus">fokusa</a> <i>F</i> i od <a href="/w/index.php?title=Direktrisa&action=edit&redlink=1" class="new" title="Direktrisa (stranica ne postoji)">direktrise</a> <i>d</i>, koja ne prolazi fokusom <i>F</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Descartesov_koordinatni_sistem">Descartesov koordinatni sistem</h4></div> <p>Standardni opis parabole: <br /> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Datoteka:Parabola_kartezsky_system.GIF" class="mw-file-description" title="Parabola u descartesovom koordinatnom sistemu"><img alt="Parabola u descartesovom koordinatnom sistemu" src="//upload.wikimedia.org/wikipedia/commons/6/68/Parabola_kartezsky_system.GIF" decoding="async" width="228" height="210" class="mw-file-element" data-file-width="228" data-file-height="210" /></a><figcaption>Parabola u descartesovom koordinatnom sistemu </figcaption></figure> <div> <p><b>V[m, n]</b> – vrh parabole sa koordinatama m, n <br /> <b>F</b> – fokus parabole <br /> <b>d</b> – direktrisa <br /> <b>o</b> – osa parabole <br /> <b>|DF| = p</b> – veličina <a href="/w/index.php?title=Parametr_(matematika)&action=edit&redlink=1" class="new" title="Parametr (matematika) (stranica ne postoji)">parametra</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da906e37e0b7cf9f266d3c9cf76f1bdf50412e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:5.907ex; height:2.509ex;" alt="{\displaystyle p>0\,\!}"></span><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |DV|=|FV|={p \over 2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |DV|=|FV|={p \over 2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df63b658d7ff05e94eea69526dcc986b06a42066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:18.416ex; height:4.843ex;" alt="{\displaystyle |DV|=|FV|={p \over 2}\,\!}"></span><br /> <b>X[x, y]</b> – proizvoljna <a href="/wiki/Ta%C4%8Dka" class="mw-disambig" title="Tačka">tačka</a> koja pripada paraboli </p> </div><p><br /> </p><div class="mw-heading mw-heading5"><h5 id="Kanonski_oblik_jednačine"><span id="Kanonski_oblik_jedna.C4.8Dine"></span>Kanonski oblik jednačine</h5></div> <p>Kanonski (normalni) oblik jednačine parabole u normalnom položaju (osa parabole je paralelna sa osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> te za vrh parabole <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=[x_{0},y_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=[x_{0},y_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61318c65b1fb533f3a9ed6c2d4c920b35bca8ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.791ex; height:2.843ex;" alt="{\displaystyle V=[x_{0},y_{0}]}"></span>) vrijedi </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(y-y_{0})}^{2}=2p(x-x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(y-y_{0})}^{2}=2p(x-x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34eea4f8fe7017b851720ee2ad422c767c07286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.847ex; height:3.343ex;" alt="{\displaystyle {(y-y_{0})}^{2}=2p(x-x_{0})}"></span></dd></dl> <p>Za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}"></span> parabola je otvorena desno a za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51432c3f75d79064093f4adfc4297ac8b7425bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p<0}"></span> parabola je otvorena lijevo. Za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=0,y_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=0,y_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f3ce918ae41f2b1207e79f42ece12b401f34af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.133ex; height:2.509ex;" alt="{\displaystyle x_{0}=0,y_{0}=0}"></span> dobija se parabola s vrhom koordinatnom početku. </p><p>Fokus, tako zadane parabole, ima koordinate </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[x_{0}+{\frac {p}{2}},y_{0}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[x_{0}+{\frac {p}{2}},y_{0}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d089ed500c4980b1caa64c6dfae8ac1c0a9d488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.652ex; height:4.843ex;" alt="{\displaystyle \left[x_{0}+{\frac {p}{2}},y_{0}\right]}"></span></dd></dl> <p>a direktrisa je opisana jednačinom </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{0}-{\frac {p}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{0}-{\frac {p}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06fb0b46a9d393746d5176399a6dc9b6c9223344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.658ex; height:4.843ex;" alt="{\displaystyle x=x_{0}-{\frac {p}{2}}}"></span></dd></dl> <p><br /> Kanonski oblik jednačine parabole s osom u osi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> i vrhom u koordinatnom početku se može zapisati kao </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=2py}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=2py}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0148df965c4a24992dc324954da262925ebf40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.97ex; height:3.009ex;" alt="{\displaystyle x^{2}=2py}"></span></dd></dl> <p>Za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}"></span> parabola je otvorena prema gore a za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51432c3f75d79064093f4adfc4297ac8b7425bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p<0}"></span> otvorena je prema dole. </p> <div class="mw-heading mw-heading5"><h5 id="Jednačina_konusnog_presjeka"><span id="Jedna.C4.8Dina_konusnog_presjeka"></span>Jednačina konusnog presjeka</h5></div> <p>Ako u jednačini <a href="/wiki/Konusni_presjek" title="Konusni presjek">konusnog presjeka</a> uvrstimo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{11}=a_{12}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{11}=a_{12}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3390ac4f1db4a5763ae3d56aa496d5220ed98c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.571ex; height:2.509ex;" alt="{\displaystyle a_{11}=a_{12}=0}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{13}a_{22}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{13}a_{22}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc1a93a328535d4a8e4f2325f16cce16f119d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.473ex; height:2.676ex;" alt="{\displaystyle a_{13}a_{22}\neq 0}"></span>, dobijemo parabolu u normalnom položaju (osa parabole je paralelna s osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>), koja ima disektrisu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {a_{23}^{2}+a_{13}^{2}-a_{22}a_{23}}{2a_{22}a_{13}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {a_{23}^{2}+a_{13}^{2}-a_{22}a_{23}}{2a_{22}a_{13}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a3be3d12bf0356c09c188a82146ed18be7cecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.369ex; height:6.343ex;" alt="{\displaystyle x={\frac {a_{23}^{2}+a_{13}^{2}-a_{22}a_{23}}{2a_{22}a_{13}}}}"></span></dd></dl> <p>fokus ima koordinate </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\left[{\frac {a_{23}^{2}-a_{13}^{2}-a_{22}a_{33}}{2a_{22}a_{13}}},\;-{\frac {a_{23}}{a_{22}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\left[{\frac {a_{23}^{2}-a_{13}^{2}-a_{22}a_{33}}{2a_{22}a_{13}}},\;-{\frac {a_{23}}{a_{22}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7fa0c816afc8d2cba63b2cd8586cbcb49d524a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.92ex; height:7.509ex;" alt="{\displaystyle F=\left[{\frac {a_{23}^{2}-a_{13}^{2}-a_{22}a_{33}}{2a_{22}a_{13}}},\;-{\frac {a_{23}}{a_{22}}}\right]}"></span></dd></dl> <p>a koordinate vrha su </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\left[{\frac {a_{23}^{2}-a_{22}a_{33}}{2a_{22}a_{13}}},\;-{\frac {a_{23}}{a_{22}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\left[{\frac {a_{23}^{2}-a_{22}a_{33}}{2a_{22}a_{13}}},\;-{\frac {a_{23}}{a_{22}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e899cbd71c11599d09f021c19d5eacf9e59eb2ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.02ex; height:7.509ex;" alt="{\displaystyle V=\left[{\frac {a_{23}^{2}-a_{22}a_{33}}{2a_{22}a_{13}}},\;-{\frac {a_{23}}{a_{22}}}\right]}"></span></dd></dl> <p>Parametar ima vrijednost </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |p|=\left|{\frac {a_{13}}{a_{22}}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mfrac> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |p|=\left|{\frac {a_{13}}{a_{22}}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c39e39ad772f4dad71d9063db43f9d461f08188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.797ex; height:5.509ex;" alt="{\displaystyle |p|=\left|{\frac {a_{13}}{a_{22}}}\right|}"></span></dd></dl> <p><br /> Slično u slučaju <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{12}=a_{22}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{12}=a_{22}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b08290bcb22be5867fa18340721f5be13876a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.571ex; height:2.509ex;" alt="{\displaystyle a_{12}=a_{22}=0}"></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{11}a_{23}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{11}a_{23}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f61830d33f0e3727d8f88e6709c76923c236fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.473ex; height:2.676ex;" alt="{\displaystyle a_{11}a_{23}\neq 0}"></span> dobijamo parabolu u normalnom položaju (osa parabole je paralelna s osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>). Za direktrisu, fokus, vrh i parametar dobijamo </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {a_{13}^{2}+a_{23}^{2}-a_{11}a_{13}}{2a_{11}a_{23}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {a_{13}^{2}+a_{23}^{2}-a_{11}a_{13}}{2a_{11}a_{23}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1fe798e39b52d0e7ce0d968b0a0fbd25ce1d20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.195ex; height:6.343ex;" alt="{\displaystyle y={\frac {a_{13}^{2}+a_{23}^{2}-a_{11}a_{13}}{2a_{11}a_{23}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\left[-{\frac {a_{13}}{a_{11}}},\;{\frac {a_{13}^{2}-a_{23}^{2}-a_{11}a_{33}}{2a_{11}a_{23}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\left[-{\frac {a_{13}}{a_{11}}},\;{\frac {a_{13}^{2}-a_{23}^{2}-a_{11}a_{33}}{2a_{11}a_{23}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c455406cc293242001cda6c83e75f36c526a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.92ex; height:7.509ex;" alt="{\displaystyle F=\left[-{\frac {a_{13}}{a_{11}}},\;{\frac {a_{13}^{2}-a_{23}^{2}-a_{11}a_{33}}{2a_{11}a_{23}}}\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\left[-{\frac {a_{13}}{a_{11}}},\;{\frac {a_{13}^{2}-a_{11}a_{33}}{2a_{11}a_{23}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\left[-{\frac {a_{13}}{a_{11}}},\;{\frac {a_{13}^{2}-a_{11}a_{33}}{2a_{11}a_{23}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cbf0e98085367b178ca52ec6e7686dbf4bd4fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.02ex; height:7.509ex;" alt="{\displaystyle V=\left[-{\frac {a_{13}}{a_{11}}},\;{\frac {a_{13}^{2}-a_{11}a_{33}}{2a_{11}a_{23}}}\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |p|=\left|{\frac {a_{23}}{a_{11}}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mfrac> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |p|=\left|{\frac {a_{23}}{a_{11}}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3535f2dff59ac0743cd9e2fd5ff80733a9bb1843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.797ex; height:5.509ex;" alt="{\displaystyle |p|=\left|{\frac {a_{23}}{a_{11}}}\right|}"></span></dd></dl> <p><br /> Parabolu iz općeg do normalnog položaja se može prevesti <a href="/w/index.php?title=Rotacija_(geometrija)&action=edit&redlink=1" class="new" title="Rotacija (geometrija) (stranica ne postoji)">rotacijom</a> koordinatnog sistema o <a href="/wiki/Ugao" title="Ugao">ugao</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> danim izrazom </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tg} 2\alpha ={\frac {2a_{12}}{a_{11}-a_{22}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tg</mi> <mo>⁡</mo> <mn>2</mn> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tg} 2\alpha ={\frac {2a_{12}}{a_{11}-a_{22}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba35e32e2dbef9b6b165ec8a0edd207c5fe3ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.091ex; height:5.509ex;" alt="{\displaystyle \operatorname {tg} 2\alpha ={\frac {2a_{12}}{a_{11}-a_{22}}}}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Karakteristike_parabole_u_odnosu_na_njen_položaj"><span id="Karakteristike_parabole_u_odnosu_na_njen_polo.C5.BEaj"></span>Karakteristike parabole u odnosu na njen položaj</h5></div> <ul><li>Osa parabole <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span> je paralelna s osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> imajući minimum(tačka V) na osi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</li></ul> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Datoteka:Parabola_kartezsky_system_vpravo.GIF" class="mw-file-description" title="Parabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu ose x"><img alt="Parabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu ose x" src="//upload.wikimedia.org/wikipedia/commons/3/3c/Parabola_kartezsky_system_vpravo.GIF" decoding="async" width="118" height="123" class="mw-file-element" data-file-width="118" data-file-height="123" /></a><figcaption>Parabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu ose x</figcaption></figure> <dl><dd><i>Tjemena jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y-n)^{2}=2p(x-m)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y-n)^{2}=2p(x-m)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebeeab0f27c026b52a529dc6bd482898553c0ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:22.091ex; height:3.176ex;" alt="{\displaystyle (y-n)^{2}=2p(x-m)\,\!}"></span></dd></dl></dd> <dd><i>Parametarska jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={p \over 2}t^{2}+m\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={p \over 2}t^{2}+m\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a974763878e0a7337826769f0911701f9c978643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:13.595ex; height:4.843ex;" alt="{\displaystyle x={p \over 2}t^{2}+m\,\!}"></span><br /></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=pt+n\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>p</mi> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=pt+n\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9da271bbe233ebbb9c651fdcf35f816807d7fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.885ex; height:2.343ex;" alt="{\displaystyle y=pt+n\,\!}"></span></dd></dl></dd> <dd><i>Opća jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}+Ax+By+C=0,A\neq 0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>A</mi> <mo>≠</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}+Ax+By+C=0,A\neq 0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b766a8471fd788ddf7413dc05172d647f8b94c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:30.18ex; height:3.176ex;" alt="{\displaystyle y^{2}+Ax+By+C=0,A\neq 0\,\!}"></span></dd></dl></dd> <dd><i>Jednačina direktrise</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=m-{p \over 2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>m</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=m-{p \over 2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6785666320ed19d7b25a7d31e3dfd63fc63a32d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:11.702ex; height:4.843ex;" alt="{\displaystyle x=m-{p \over 2}\,\!}"></span></dd></dl></dd> <dd><i>Jednačina <a href="/w/index.php?title=Tangenta&action=edit&redlink=1" class="new" title="Tangenta (stranica ne postoji)">tangente</a> u tački <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T[x_{0},y_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[x_{0},y_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22a552326ac13d868cad93d2450a74c23c84201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.541ex; height:2.843ex;" alt="{\displaystyle T[x_{0},y_{0}]}"></span>: </i> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y-n)(y_{0}-n)=p(x+x_{0}-2m)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y-n)(y_{0}-n)=p(x+x_{0}-2m)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/839077e8057500fac3ca9c3c7f5b94ee051cf64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:34.499ex; height:2.843ex;" alt="{\displaystyle (y-n)(y_{0}-n)=p(x+x_{0}-2m)\,\!}"></span></dd></dl></dd></dl> <p>Osa parabole <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span> je paralelna s osoom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> imajući maximum(tačka V) na osi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.<br /> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Datoteka:Parabola_kartezsky_system_vlevo.GIF" class="mw-file-description" title="Parabola u Descartesovom koordinatnom sistemu usmjerena ka negativnom djelu ose x"><img alt="Parabola u Descartesovom koordinatnom sistemu usmjerena ka negativnom djelu ose x" src="//upload.wikimedia.org/wikipedia/commons/0/0b/Parabola_kartezsky_system_vlevo.GIF" decoding="async" width="118" height="123" class="mw-file-element" data-file-width="118" data-file-height="123" /></a><figcaption>Parabola u Descartesovom koordinatnom sistemu usmjerena ka negativnom djelu ose <i>x</i></figcaption></figure> <dl><dd><i>Tjemena jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y-n)^{2}=-2p(x-m)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y-n)^{2}=-2p(x-m)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3902285c07dfee91f0e379e2fd660b415e867fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:23.899ex; height:3.176ex;" alt="{\displaystyle (y-n)^{2}=-2p(x-m)\,\!}"></span></dd></dl></dd> <dd><i>Parametarska jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-{p \over 2}t^{2}+m\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-{p \over 2}t^{2}+m\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb63a8f80fb6f0fcd9b1e412f9738163589e7118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:15.404ex; height:4.843ex;" alt="{\displaystyle x=-{p \over 2}t^{2}+m\,\!}"></span><br /></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=-pt+n\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=-pt+n\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0700329599f7e611477d4d5630c1bec53105504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:12.693ex; height:2.343ex;" alt="{\displaystyle y=-pt+n\,\!}"></span></dd></dl></dd> <dd><i>Opća jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}+Ax+By+C=0,A\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>A</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}+Ax+By+C=0,A\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b632739740d6fcb946b624ee284b5349eca5c26e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.793ex; height:3.176ex;" alt="{\displaystyle y^{2}+Ax+By+C=0,A\neq 0}"></span></dd></dl></dd> <dd><i>Jenačina direktrise</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=m+{p \over 2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=m+{p \over 2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad401b817eab7d3cac060f36655d1b0c9b8a93b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:11.702ex; height:4.843ex;" alt="{\displaystyle x=m+{p \over 2}\,\!}"></span></dd></dl></dd> <dd><i>Jednačina <a href="/w/index.php?title=Tangenta&action=edit&redlink=1" class="new" title="Tangenta (stranica ne postoji)">tangente</a> u <a href="/wiki/Ta%C4%8Dka" class="mw-disambig" title="Tačka">tački</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T[x_{0},y_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[x_{0},y_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22a552326ac13d868cad93d2450a74c23c84201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.541ex; height:2.843ex;" alt="{\displaystyle T[x_{0},y_{0}]}"></span></i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y-n)(y_{0}-n)=-p(x+x_{0}-2m)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (y-n)(y_{0}-n)=-p(x+x_{0}-2m)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df483f2cd72aff4063d7e35ed4524cf921415673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:36.307ex; height:2.843ex;" alt="{\displaystyle (y-n)(y_{0}-n)=-p(x+x_{0}-2m)\,\!}"></span></dd></dl></dd></dl> <ul><li>Osa parabole <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span> je paralelna s osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> imajući minimum. Konveksna parabola.</li></ul> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Datoteka:Parabola_kartezsky_system_nahore.GIF" class="mw-file-description" title="Parabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu y"><img alt="Parabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu y" src="//upload.wikimedia.org/wikipedia/commons/1/15/Parabola_kartezsky_system_nahore.GIF" decoding="async" width="118" height="123" class="mw-file-element" data-file-width="118" data-file-height="123" /></a><figcaption>Parabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu <i>y</i></figcaption></figure> <dl><dd><i>Tjemena jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-m)^{2}=2p(y-n)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-m)^{2}=2p(y-n)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ff2c815afd3be94341a2f1b8cfb3a9abd2dda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:22.091ex; height:3.176ex;" alt="{\displaystyle (x-m)^{2}=2p(y-n)\,\!}"></span></dd></dl></dd> <dd><i>Parametarska jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=pt+m\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>p</mi> <mi>t</mi> <mo>+</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=pt+m\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e5da4fc872039ee4d2b0822370d32647895150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:11.705ex; height:2.343ex;" alt="{\displaystyle x=pt+m\,\!}"></span><br /></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={p \over 2}t^{2}+n\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={p \over 2}t^{2}+n\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384c219d0f0198148bd8ab05a302c4156e80e444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:12.776ex; height:4.843ex;" alt="{\displaystyle y={p \over 2}t^{2}+n\,\!}"></span></dd></dl></dd> <dd><i>Opća jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+Ax+By+C=0,B\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>B</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+Ax+By+C=0,B\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95201df7e544be94c585d185bbbd306a11e110c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.984ex; height:3.176ex;" alt="{\displaystyle x^{2}+Ax+By+C=0,B\neq 0}"></span></dd></dl></dd> <dd><i>Jednačina direktrise</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=n-{p \over 2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=n-{p \over 2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a56b12ac75719209ad8ff469fc75542d278f3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:10.882ex; height:4.843ex;" alt="{\displaystyle y=n-{p \over 2}\,\!}"></span></dd></dl></dd> <dd><i>Jednačina tangente u tački <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T[x_{0},y_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[x_{0},y_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22a552326ac13d868cad93d2450a74c23c84201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.541ex; height:2.843ex;" alt="{\displaystyle T[x_{0},y_{0}]}"></span></i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-m)(x_{0}-m)=p(y+y_{0}-2n)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-m)(x_{0}-m)=p(y+y_{0}-2n)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f39204dec1c1a743f3f709374d64d5563ae3e5dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:35.145ex; height:2.843ex;" alt="{\displaystyle (x-m)(x_{0}-m)=p(y+y_{0}-2n)\,\!}"></span></dd></dl></dd></dl> <ul><li>Osa parabole <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1031f61947aa3d1cf3a70ec3e4904df2c3675d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle o}"></span> je paralelna s osom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> imajući maksimum. Konkavna parabola.</li></ul> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/Datoteka:Parabola_kartezsky_system_dole.GIF" class="mw-file-description" title="desnoParabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu y"><img alt="desnoParabola u Descartesovom koordinatnom sistemu usmjerena ka pozitivnom djelu y" src="//upload.wikimedia.org/wikipedia/commons/b/b3/Parabola_kartezsky_system_dole.GIF" decoding="async" width="118" height="123" class="mw-file-element" data-file-width="118" data-file-height="123" /></a></span> </p> <dl><dd><i>Tjemena jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-m)^{2}=-2p(y-n)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-m)^{2}=-2p(y-n)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1256497f0d7d5af2fb18e879fbcc6a43a2df516e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:23.899ex; height:3.176ex;" alt="{\displaystyle (x-m)^{2}=-2p(y-n)\,\!}"></span></dd></dl></dd> <dd><i>Parametarska jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-pt+m\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mi>t</mi> <mo>+</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-pt+m\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2431529240bb9e166ff486d783a3485d28e404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:13.513ex; height:2.343ex;" alt="{\displaystyle x=-pt+m\,\!}"></span><br /></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=-{p \over 2}t^{2}+n\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=-{p \over 2}t^{2}+n\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0c260cdc70a55209c3e86a18cc7001b43bbd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:14.584ex; height:4.843ex;" alt="{\displaystyle y=-{p \over 2}t^{2}+n\,\!}"></span></dd></dl></dd> <dd><i>Opća jednačina</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+Ax+By+C=0,B\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>B</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+Ax+By+C=0,B\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95201df7e544be94c585d185bbbd306a11e110c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.984ex; height:3.176ex;" alt="{\displaystyle x^{2}+Ax+By+C=0,B\neq 0}"></span></dd></dl></dd> <dd><i>Jednačina direktrise</i>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=n+{p \over 2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=n+{p \over 2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea82103d91c88c962c9dee66e3d87ffa82bec883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:10.882ex; height:4.843ex;" alt="{\displaystyle y=n+{p \over 2}\,\!}"></span></dd></dl></dd> <dd><i>Jednačina <a href="/w/index.php?title=Tangenta&action=edit&redlink=1" class="new" title="Tangenta (stranica ne postoji)">tangente</a> u tački <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T[x_{0},y_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T[x_{0},y_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22a552326ac13d868cad93d2450a74c23c84201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.541ex; height:2.843ex;" alt="{\displaystyle T[x_{0},y_{0}]}"></span>: </i> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-m)(x_{0}-m)=-p(y+y_{0}-2n)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-m)(x_{0}-m)=-p(y+y_{0}-2n)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83128ad491e1339d64dc0b178aa3fef492dceb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:36.953ex; height:2.843ex;" alt="{\displaystyle (x-m)(x_{0}-m)=-p(y+y_{0}-2n)\,\!}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Uzajamni_odnos_parabole_i_prave">Uzajamni odnos parabole i prave</h5></div> <p>Rješimo <a href="/w/index.php?title=Sistem_jedna%C4%8Dina&action=edit&redlink=1" class="new" title="Sistem jednačina (stranica ne postoji)">sistem jednačina</a> parabole i <a href="/w/index.php?title=Prava&action=edit&redlink=1" class="new" title="Prava (stranica ne postoji)">prave</a>. Ukoliko dobijemo linearnu jednačinu, koja ima rješenja - prava sječe parabolu u jednoj tački. Ukoliko linearna jednačina nema rješenja - prava i parabola se mimoilaze. Ukoliko dobijemo <a href="/wiki/Kvadratna_jedna%C4%8Dina" title="Kvadratna jednačina">kvadratnu jednačinu</a> i <a href="/w/index.php?title=Diskriminanta&action=edit&redlink=1" class="new" title="Diskriminanta (stranica ne postoji)">diskriminanta</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> je: </p> <ul><li>D > 0 dva rješenja - prava sječe parabolu u dvije tačke</li> <li>D = 0 jedno rješenje - prava je paraboli tangenta</li> <li>D < 0 nema rješenja - prava i parabola se mimoilaze</li></ul> <div class="mw-heading mw-heading4"><h4 id="Polarni_koordinatni_sistem">Polarni koordinatni sistem</h4></div> <p>Parabola s fokusom u početku koordinatnog sistema i s vrhom na negativnoj poluosi x zapisuje se pomoću jednačine: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(1-\cos \varphi )=p\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(1-\cos \varphi )=p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be216e509358fdf0e6c76f5bc2a17c1fa264a5fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.534ex; height:2.843ex;" alt="{\displaystyle r(1-\cos \varphi )=p\,}"></span></dd></dl> <p>gdje <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}"></span> je parameter parabole. </p><p>Iz tog je vidljivo, da parametar parabole ima također značenje polovine dužine tzv. <a href="https://en.wikipedia.org/wiki/latus_rectum" class="extiw" title="en:latus rectum">latus rectum</a>, tako da je i <a href="/wiki/Tetiva_(geometrija)" title="Tetiva (geometrija)">tetiva</a> konusnog presjeka okomita na glavnu osu u fokusu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>. Kod parabole se ta vrijednost izjednačava sa četverostrukom dužinom fokusne udaljenosti. </p><p>Polarnom jednačinom je moguće dokazati, da parabola nastane kružnom inverzijom <a href="/w/index.php?title=Kardioda&action=edit&redlink=1" class="new" title="Kardioda (stranica ne postoji)">karadiode</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Parabola_u_relnom_svijetu">Parabola u relnom svijetu</h2></div> <p><a href="/w/index.php?title=Trajektorija&action=edit&redlink=1" class="new" title="Trajektorija (stranica ne postoji)">Trajektorije</a> <a href="/wiki/Tijelo" class="mw-redirect" title="Tijelo">tijela</a> koja se kreću u homogenom <a href="/wiki/Gravitacija" title="Gravitacija">gravitacijskom polju</a> je parabola. Po <b>paraboli</b> se također kreću tijela u centralnim gravitacijskim poljima, ako je njihova <a href="/wiki/Brzina" title="Brzina">brzina</a> tačno jednaka drugoj kosmičkoj brzini a smjer im se poklapa sa smjerom tog polja. Npr. <a href="/wiki/Put" title="Put">put</a>, po kojem se kreću neke <a href="/wiki/Kometa" title="Kometa">kometi</a>, su veoma slične paraboli. </p><p>Ako se <a href="/w/index.php?title=Zraka&action=edit&redlink=1" class="new" title="Zraka (stranica ne postoji)">zraka</a> koja prilazi paraboli (ili <a href="/wiki/Paraboloid" title="Paraboloid">paraboloidu</a>) paralelno sa osom simetrije odbije od parabole/paraboloida, prolazit će fokusom. To je razlog, zašto se proizvode <a href="/w/index.php?title=Paraboli%C4%8Dko_ogledalo&action=edit&redlink=1" class="new" title="Paraboličko ogledalo (stranica ne postoji)">parabolička ogledala</a> i <a href="/wiki/Antena" title="Antena">antene</a> (npr. kod <a href="/wiki/Automobil" title="Automobil">automobila</a>, <a href="/w/index.php?title=Dvogled&action=edit&redlink=1" class="new" title="Dvogled (stranica ne postoji)">dvogleda</a>, <a href="/wiki/Satelit" class="mw-redirect" title="Satelit">telekomunikacijskih satelita</a> i sl.). </p> <div class="mw-heading mw-heading2"><h2 id="Također_pogledajte"><span id="Tako.C4.91er_pogledajte"></span>Također pogledajte</h2></div> <ul><li><a href="/wiki/Kru%C5%BEnica" title="Kružnica">Kružnica</a></li> <li><a href="/wiki/Hiperbola" title="Hiperbola">Hiperbola</a></li> <li><a href="/wiki/Konusni_presjek" title="Konusni presjek">Konusni presjek</a></li> <li><a href="/wiki/Elipsa" title="Elipsa">Elipsa</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vanjski_linkovi">Vanjski linkovi</h2></div> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Parabola.html">Parabola u enciklopediji MathWorld</a> (engleski)</li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Preuzeto iz "<a dir="ltr" href="https://bs.wikipedia.org/w/index.php?title=Parabola_(matematika)&oldid=1722722">https://bs.wikipedia.org/w/index.php?title=Parabola_(matematika)&oldid=1722722</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorija</a>: <ul><li><a href="/wiki/Kategorija:Matematika" title="Kategorija:Matematika">Matematika</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Ova stranica je posljednji put izmijenjena na datum 10 mart 2012 u 00:00.</li> <li id="footer-info-copyright">Tekst je dostupan pod <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/">slobodnom licencom Autorstvo-Dijeliti pod istim uvjetima</a>; mogu se primijeniti i dodatni uvjeti. 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