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 Statistikk ChatOm 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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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         \u003cp class=\"noindent\" \u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eLille halvakse:\u003c/span\u003e \u003c/p\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         \u003cp class=\"noindent\" \u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eStore halvakse:\u003c/span\u003e \u003c/p\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\"\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\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" 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      \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\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\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\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\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\" \u003e\u003cmtable  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\"\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\"\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\"\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\"\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/b\u003e                                                                                               \u003c/div\u003e  \u003c/div\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\" \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\"\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\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\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\u003c/p\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  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\"\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\"\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\"\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  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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\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  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                        \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\"\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\"\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\"\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\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\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\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\"\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\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\"\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     "},"isRootPath":false,"otherLocales":[],"type":"encyclopedia","h1":"Likningen til en ellipse","siblings":[{"h1":"Ellipser","path":"/encyclopedia/geometri/figurer-i-planet/sirkler/ellipse/ellipser"},{"h1":"Likningen til en ellipse","path":"/encyclopedia/geometri/figurer-i-planet/sirkler/ellipse/likningen-til-en-ellipse"}],"metaDescription":"Lær om likningen for ellipse. 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Kom i gang ved å ta quizen!","quizMessageTitle":"Kartleggingsquiz","youHaveNews":"Du har nyheter i basen! \u003copenHomebase\u003eÅpne basen\u003c/openHomebase\u003e"},"chatBetweenUserAndTutor":"Chat mellom bruker og privatlærer","imageAlt":{"background":"Bønnestengel","backgroundMobileTop":"Bønnestengel","backgroundMobileBottom":"Bønnestengel","homebasePlaceholder":"Dragebaby som titter ut av et egg"},"featureProducts":{"label":"Hovedemner i matte"},"statistics":{"label":"Min statistikk","activity":{"label":"Totalt antall aktiviteter","correct":"Riktig","inCorrect":"Galt","instruction":"Prøv produktene våre for å låse opp denne statistikken"},"timeSpent":{"label":"Tid brukt","days":"Dager","hours":"Timer","minutes":"Minutter"},"feedback":{"yes":"Ja","no":"Nei","instruction":"Ønsker du å se mer statistikk i fremtiden?"}},"checklists":{"title":"Kom i gang","description":"Fullfør sjekklisten og bli belønnet.","seeMore":"Se mer","close":"Lukk","onboarding":"Fullfør mapping quiz","createAvatar":"Lag avatar","exploreSpace":"Besøk Space","mathTask":"Fullfør et matteoppdrag","exploreLeaderboard":"Besøk topplisten","verifyEmail":"Verifiser e-post","completeProfile":"Fullfør profilen din","mathTaskBadge":"Bekjemp Bootcamp-bossen","tooltip":{"onboarding":"\u003cstrong\u003eTjen {{rewardXP}}{{comma}}{{rewardGems}}\u003c/strong\u003e ved å fullføre \u003cstrong\u003emapping quiz.\u003c/strong\u003e","createAvatar":"\u003cstrong\u003eLag\u003c/strong\u003e din egen avatar og \u003cstrong\u003etjen {{rewardXP}}{{comma}}{{rewardGems}}.\u003c/strong\u003e","exploreSpace":"\u003cstrong\u003eTjen {{rewardXP}}{{comma}}{{rewardGems}}\u003c/strong\u003e ved å utforske \u003cstrong\u003eSpace\u003c/strong\u003e","mathTask":"\u003cstrong\u003eFullfør\u003c/strong\u003e ditt første matteoppdrag ved å bekjempe \u003cstrong\u003eBootcamp-bossen.\u003c/strong\u003e","exploreLeaderboard":"\u003cstrong\u003eBesøk\u003c/strong\u003e vår globale toppliste \u003cstrong\u003etjen {{rewardXP}}{{comma}}{{rewardGems}}!\u003c/strong\u003e","verifyEmail":"\u003cstrong\u003eVerifiser\u003c/strong\u003e e-posten din og tjen \u003cstrong\u003e{{rewardXP}}{{comma}}{{rewardGems}}.\u003c/strong\u003e","completeProfile":"\u003cstrong\u003eFullfør\u003c/strong\u003e profilen din og tjen \u003cstrong\u003e{{rewardXP}}{{comma}}{{rewardGems}}.\u003c/strong\u003e","rewadXP":"{{XP}} XP","rewadGems":"{{GEMS}} gems"},"notification":{"onboarding":"Du har fullført mapping quizen.","createAvatar":"Du har laget en avatar.","exploreSpace":"Du har besøkt Space.","mathTask":"Du have fullført ditt første matteoppdrag.","exploreLeaderboard":"Du har besøkt den globale topplisten.","verifyEmail":"Du har verifisert e-posten din.","completeProfile":"Du har fullført profilen din."},"button":{"onboarding":"Mapping Quiz","createAvatar":"Lag avatar","exploreSpace":"Gå til Space","mathTask":"Mine matteoppdrag","exploreLeaderboard":"Topplisten","verifyEmail":"Verifiser e-post","completeProfile":"Fullfør profilen din","claimReward":"Motta belønning"},"imageAlt":{"onboarding":"Hjerne-ikon","createAvatar":"Avatar-ikon","exploreSpace":"Space-ikon","mathTask":"Matteoppdrag-ikon","exploreLeaderboard":"Trofé-ikon","verifyEmail":"E-mail-ikon","completeProfile":"Profil-ikon","notification_checkbox":"Sjekkliste-notifikasjon sjekkboks"},"checklist_item_completed":"Sjekklistepunkt fullført"},"completed":"Fullført","reward":"Belønning","exploreMathTopics":"Utforsk hovedemner i matte","completeLevel":"Fullfør \u003cb\u003enivå {{level}}\u003c/b\u003e for å låse opp","gotIt":"Skjønner","lockedChest":"Kisten er låst","congratulations":"Gratulerer!","collectRewards":"Du har fått \u003cb\u003eKudos\u003c/b\u003e og \u003cb\u003ekrystaller\u003c/b\u003e som du kan bruke i din personlige \u003cb\u003eSpace!\u003c/b\u003e ","collect":"Motta","bossBattle":"Boss Battle","challenge":"Utfordre bossen","goToVideo":"Gå til videoen","goToActivity":"Gå til aktiviteten","goToBoss":"Gå til bossen","startHere":"Start her","read":"Les","goToArticle":"Gå til artikellen","skip":"Hopp frem","completeTask":"Fullfør matteoppdrag","task":"Oppdrag","readyProgress":"Klar til å gå videre?","defeatBoss":"\u003cb\u003eBekjemp\u003c/b\u003e bossen for å \u003cb\u003efullføre\u003c/b\u003e matteoppdraget!"},"gamification":{"questSummary":{"pointsEarned":"Opptjente poeng","reward":"Trofé","starsEarned":"Opptjente stjerner","title":"Sammendrag"},"globalLeaderboard":"Global toppliste","title":"Toppliste","weekNo":"Uke {{ weekNo }}","rank":"Plass","youEarned":"Du tjente","congratulations":"Gratulerer","bravo":"Bravo!","ohNo":"Oh nei!","youPromoted":"Du gikk opp til","youDemoted":"Du gikk ned til","youStayed":"Du beholdt plassen din i","youJoined":"Du er med i \u003cstrong\u003e{{league}}\u003c/strong\u003e denne uken","league":{"bronze":"bronseligaen","silver":"sølvligaen","gold":"gull-ligaen","platinum":"platinumligaen","diamond":"diamantligaen"},"yourRank":"med \u003cstrong\u003e{{rank}}.\u003c/strong\u003e plass! 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Karakterene mine har forbedret seg drastisk.","lakshinAge":"9 år gammel.","lakshinSriLankaTestimonial":"Siden tilbyr spennende videoer og øvelser som jeg enkelt kan huske.","aleksanderAge":"32 år.","aleksanderTestimonial":"Det har vært et flott verktøy og en fin måte for noen i min situasjon å gå tilbake og forbedre sine matematikkferdigheter fra grunnen av.","bridgetteAge":"53 år.","bridgetteTestimonial":"Jeg kan nå hjelpe min 12 år gamle sønn med lekser. Jeg har også fått bedre selvtillit på jobben","dianneAge":"36 år.","dianneTestimonial":"Siden hjelper meg å lære matematikk slik at jeg kan hjelpe barna mine med leksene deres.","avhatakaliAge":"16 år gammel.","avhatakaliTestimonial":"Å ha en matematikk-hjelper jeg kan øve sammen med gjør at jeg forbedrer meg mye raskere enn når jeg øver alene.","edgardoAge":"72 år gammel.","edgardoTestimonial":"Hyppig øving på siden har forbedret mine mattekunnskaper, og det er på grunn av enkle og klare forklaringer. 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