Likningen til en ellipse

Ellipse med brennpunkt, stor- og liten halvakse.

Teori

Ellipsen

En ellipse består av alle punktene som har lik samlet avstand til to gitte punkter, kalt brennpunkter.

Følgende lengder er viktige for å beskrive en ellipse:

Lille halvakse:

den korteste avstanden fra origo til et punkt på ellipsen.

Store halvakse:

den lengste avstanden fra origo til et punkt på ellipsen.

Regel

Likningen for en ellipse

Dersom en ellipse har brennpunkter som ligger på $x$ -aksen med lik avstand til origo, og du kaller store halvakse $a$ og lille halvakse $b$ , er likningen for ellipsen

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 .

En sirkel er et spesialtilfelle av en ellipse, nemlig tilfellet der brennpunktene ligger oppå hverandre.

Eksempel 1

Finn likningen for en ellipse der lengden til store halvakse er $a = 5$ og lengden til lille halvakse er $b = 3$

\begin{array}{l} x 2 5^{2} + \frac{y^{2}}{3^{2}} = 1 \frac{x^{2}}{25} + \frac{y^{2}}{9} & = 1Eksempel 2 Vis at 9 x^{2} + 4 y^{2} - 36 = 0 er likningen for en ellipse For å vise at dette er en ellipse, må du gjøre om uttrykket slik at det er på formen til ellipselikningen. Denne finner du ved å gjøre om på uttrykket: \begin{array}{l} 9 x^{2} + 4 y^{2} - 36 & = 0 \\ 9 x^{2} + 4 y^{2} & = 36 & | : 9 \\ x^{2} + \frac{4 y^{2}}{9} & = 4 & | : 4 \\ \frac{x^{2}}{4} + \frac{y^{2}}{9} & = 1 \end{array} Siden du har klart å gjøre om uttrykket til en ellipselikning så har du vist at 9 x^{2} + 4 y^{2} - 36 = 0 er en ellipse, med lille halvakse a = \sqrt{4} = 2 og store halvakse b = \sqrt{9} = 3 .Vil du vite mer? Registrer deg Det er gratis!Forrige oppslag EllipserTilbakeHjem All matte Spill Progresjon LigaOm oss Om House of Math Om ansatte Karriere Media Foredrag Blogg Kontakt support@houseofmath.com Booking av privatundervisning +47 22 150 300 Adresse Bygdøy allé 23
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src=\"https://images.houseofmath.com/overleaf-assets/images/ellipse.svg\" alt=\"Ellipse med brennpunkt, stor- og liten halvakse.\" width=\"451\" height=\"451\"\u003e\u003c/p\u003e\u003c/div\u003e      \u003cdiv class=\"tcolorbox fooinfo\" id=\"tcolobox-762\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Teori\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"info-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eEllipsen\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eEn \u003ca href=\"/no/encyclopedia/geometri/figurer-i-planet/sirkler/ellipse\"\u003eellipse\u003c/a\u003e består av alle \u003ca href=\"/no/encyclopedia/geometri/figurer-i-planet/konsepter/geometriske-steder\"\u003epunktene\u003c/a\u003e som har lik samlet avstand til to gitte punkter, kalt \u003cspan  class=\"ec-lmri-12x-x-144\"\u003ebrennpunkter\u003c/span type=\"punct\"\u003e. \u003c/p\u003e\u003cp class=\"noindent\" \u003eFølgende lengder er viktige for å beskrive en ellipse:         \u003c/p\u003e\u003cdl class=\"description\"\u003e\u003cdt class=\"description\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eLille halvakse:\u003c/span\u003e \u003c/dt\u003e\u003cdd  class=\"description\"\u003e       \u003cp class=\"noindent\" \u003eden         korteste         avstanden         fra         origo         til         et         punkt         på         ellipsen.         \u003c/p\u003e\u003c/dd\u003e\u003cdt class=\"description\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eStore halvakse:\u003c/span\u003e \u003c/dt\u003e\u003cdd  class=\"description\"\u003e       \u003cp class=\"noindent\" \u003eden         lengste         avstanden         fra         origo         til         et         punkt         på         ellipsen. \u003c/p\u003e         \u003cp class=\"noindent\" \u003e\u003c/dd\u003e\u003c/dl\u003e \u003c/p\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                  \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-763\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Regel\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eLikningen\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003efor\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003een\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eellipse\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eDersom en ellipse har brennpunkter som ligger på \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e-aksen med lik avstand til origo, og du kaller store halvakse \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e og lille halvakse \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math\u003e, er likningen for ellipsen \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                               \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"punct\"\u003e.\u003c/mo\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                             \u003cp class=\"indent\" \u003e     En sirkel er et spesialtilfelle av en ellipse, nemlig tilfellet der brennpunktene ligger oppå hverandre. \u003c/p\u003e \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-764\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Eksempel 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eFinn\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003elikningen\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003efor\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003een\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eellipse\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eder\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003elengden\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003etil\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003estore\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ehalvakse\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eer\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eog\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003elengden\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003etil\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003elille\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ehalvakse\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eer\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" mathvariant=\"bold-italic\"\u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                                     \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mtd\u003e                                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e \u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e    \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                         \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-765\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Eksempel 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eVis\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eat\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eer\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003elikningen\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003efor\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003een\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eellipse\u003c/span\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eFor å vise at dette er en ellipse, må du gjøre om uttrykket slik at det er på formen til ellipselikningen. Denne finner du ved å gjøre om på uttrykket: \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"alignat-star\"\u003e                                      \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e                                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e                                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e    \u003cmrow  \u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003c/mtd\u003e                                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e                                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003c/mtd\u003e                                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003eSiden du har klart å gjøre om uttrykket til en ellipselikning så har du vist at \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                           \u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e \u003cp class=\"noindent\" \u003eer en ellipse, med lille halvakse \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/math\u003e og store halvakse \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e                                                                                                                            \u003cp class=\"indent\" \u003e     "},"sharedLanguageKey":null,"metaDescription":"Lær om likningen for ellipse. 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