A f\u00f3rmula para a equa\u00e7\u00e3o da elipse<\/strong> em coordenadas cartesianas \u00e9:<\/p>\n<\/p>\n Ouro:<\/p>\n E<\/p>\n s\u00e3o as coordenadas do centro da elipse:<\/p>\n \u00e9 o raio horizontal da elipse.<\/li>\n \u00e9 o raio vertical da elipse. <\/li>\n<\/ul>\n<\/div>\n Um tipo de elipse muito comum \u00e9 aquela cujo centro est\u00e1 na origem das coordenadas, ou seja, no ponto (0,0). \u00c9 por isso que veremos como determinar a equa\u00e7\u00e3o da elipse centrada na origem.<\/p>\n Seguindo a f\u00f3rmula para a equa\u00e7\u00e3o da elipse:<\/p>\n<\/p>\n Se a elipse estiver centrada na origem das coordenadas, isso significa que<\/p>\n<\/p>\n E<\/p>\n s\u00e3o iguais a 0, ent\u00e3o sua equa\u00e7\u00e3o ser\u00e1:<\/p>\n<\/p>\n H\u00e1 matem\u00e1ticos que tamb\u00e9m chamam essa express\u00e3o de equa\u00e7\u00e3o can\u00f4nica ou equa\u00e7\u00e3o reduzida da elipse.<\/p>\n Depois de vermos como \u00e9 a equa\u00e7\u00e3o da elipse, veremos quais s\u00e3o seus elementos. Mas primeiro, vamos lembrar o que \u00e9 exatamente uma elipse:<\/p>\n A elipse \u00e9 uma linha plana, fechada e curva muito semelhante \u00e0 circunfer\u00eancia, mas seu formato \u00e9 mais oval. Em particular, a elipse \u00e9 o lugar geom\u00e9trico de todos os pontos de um plano cuja soma das dist\u00e2ncias a dois outros pontos fixos (chamados focos F e F’) \u00e9 constante.<\/p>\n Portanto, os elementos de uma elipse s\u00e3o:<\/p>\n Os diferentes elementos de uma elipse est\u00e3o interligados. Al\u00e9m disso, as rela\u00e7\u00f5es entre eles s\u00e3o muito importantes para exerc\u00edcios sobre elipses, pois geralmente s\u00e3o necess\u00e1rias para resolver problemas sobre elipses e determinar suas equa\u00e7\u00f5es.<\/p>\n Como vimos acima na defini\u00e7\u00e3o da elipse, a dist\u00e2ncia de qualquer ponto da elipse ao foco F mais a dist\u00e2ncia do mesmo ponto ao foco F’ \u00e9 constante. Bem, esse valor constante \u00e9 igual a duas vezes o que o semieixo maior mede. Em outras palavras, a seguinte igualdade \u00e9 v\u00e1lida para qualquer ponto de uma elipse:<\/p>\n<\/p>\n Ouro<\/p>\n<\/p>\n E<\/p>\n \u00e9 a dist\u00e2ncia do ponto P ao foco F e F’ respectivamente e<\/p>\n \u00e9 o comprimento do eixo semifocal.<\/p>\n Portanto, como o v\u00e9rtice do eixo secund\u00e1rio est\u00e1 exatamente no meio do eixo focal, a dist\u00e2ncia dele a um dos focos \u00e9 equivalente ao comprimento do eixo semiprim\u00e1rio (<\/p>\n<\/p>\n ): <\/p>\n Assim, a partir do teorema de Pit\u00e1goras<\/a> , \u00e9 poss\u00edvel encontrar a rela\u00e7\u00e3o que existe entre o semieixo principal, o semieixo secund\u00e1rio e a meia dist\u00e2ncia focal:<\/strong><\/p>\n<\/p>\n Lembre-se desta f\u00f3rmula porque ser\u00e1 muito \u00fatil para calcular os resultados de exerc\u00edcios com elipses. <\/p>\n Obviamente, nem todas as elipses s\u00e3o iguais, mas algumas s\u00e3o mais alongadas e outras mais achatadas. Portanto, existe um coeficiente que serve para medir o qu\u00e3o arredondada \u00e9 uma determinada elipse. Este coeficiente \u00e9 denominado excentricidade<\/strong> e \u00e9 calculado com a seguinte f\u00f3rmula:<\/p>\n<\/p>\n Ouro<\/p>\n<\/p>\n \u00e9 a dist\u00e2ncia do centro da elipse a um de seus focos e<\/p>\n o comprimento do semieixo maior. <\/p>\n Como voc\u00ea pode ver na representa\u00e7\u00e3o anterior, quanto menor o valor da excentricidade da elipse, mais ela se assemelha a um c\u00edrculo, por outro lado, quanto maior o coeficiente, mais achatada \u00e9 a elipse. Al\u00e9m disso, o valor da excentricidade varia de zero (c\u00edrculo perfeito) a um (linha horizontal), ambos n\u00e3o inclusivos.<\/p>\n<\/p>\n Une fois que nous avons vu toutes les propri\u00e9t\u00e9s de l’ellipse, nous allons r\u00e9soudre un probl\u00e8me d’ellipse \u00e0 titre d’exemple :<\/p>\n Pour d\u00e9terminer l’\u00e9quation d’une ellipse, nous avons besoin de la longueur du demi-axe principal, de la longueur du demi-axe secondaire et des coordonn\u00e9es de son point.<\/strong> Par cons\u00e9quent, dans ce cas, nous n’avons besoin de conna\u00eetre que l’axe semi-secondaire. Ainsi, pour calculer la longueur mesur\u00e9e par l’axe semi-secondaire, nous pouvons utiliser la relation entre l’axe semi-principal, l’axe semi-secondaire et la distance semi-focale : ” title=”Rendered by QuickLaTeX.com” height=”215″ width=”2133″ style=”vertical-align: -5px;”><\/p>\n a^2=b^2+c^2 b^2=a^2-c^2 b=\\sqrt{a^2-c^2} = \\sqrt{5^2-4^2}=\\sqrt {9} = 3<\/p>\n \\cfrac{(x-x_0)^2}{a^2}+\\cfrac{(y-y_0)^2}{b^2} = 1\\cfrac{(x-4)^2}{5^2 }+\\cfrac{(y-(-1))^2}{3^2} = 1\\cfrac{\\bm{(x-4)^2}}{\\bm{25}}+\\cfrac{\\ bm{(y+1)^2}}{\\bm{9}} \\bm{= 1}<\/p>\n Quelle est l’\u00e9quation de l’ellipse centr\u00e9e au point C(2,0) dont l’axe semi-principal (parall\u00e8le \u00e0 l’axe X) et l’axe secondaire mesurent respectivement 6 et 3 unit\u00e9s ? Repr\u00e9senter graphiquement ladite ellipse. <\/p>\n L’\u00e9quation de l’ellipse est la suivante :” title=”Rendered by QuickLaTeX.com” height=”208″ width=”1595″ style=”vertical-align: -20px;”><\/p>\n \\cfrac{(x-x_0)^2}{a^2}+\\cfrac{(y-y_0)^2}{b^2} = 1<\/p>\n \\cfrac{(x-2)^2}{6^2}+\\cfrac{(y-0)^2}{3^2} = 1\\cfrac{\\bm{(x-2)^2}} {\\bm{36}}+\\cfrac{\\bm{y^2}}{\\bm{9}} \\bm{= 1}<\/p>\n Calculer l’\u00e9quation de l’ellipse dont le demi-axe principal (parall\u00e8le \u00e0 l’axe des abscisses) mesure 13 unit\u00e9s, son centre est l’origine des coordonn\u00e9es et la distance de son centre \u00e0 l’un de ses foyers est de 5 unit\u00e9s. <\/p>\n Pour calculer l’\u00e9quation de l’ellipse, nous devons savoir combien de temps mesure l’axe semi-secondaire. Et, pour cela, on peut utiliser la relation math\u00e9matique qui existe entre le demi-axe principal, le demi-axe secondaire et la demi-distance focale : ” title=”Rendered by QuickLaTeX.com” height=”299″ width=”2688″ style=”vertical-align: -20px;”><\/p>\n a^2=b^2+c^2 b^2=a^2-c^2 b=\\sqrt{a^2-c^2} = \\sqrt{13^2-5^2}=\\sqrt {144} = 12<\/p>\n \\cfrac{(x-x_0)^2}{a^2}+\\cfrac{(y-y_0)^2}{b^2} = 1\\cfrac{(x-0)^2}{13^2 }+\\cfrac{(y-0)^2}{12^2} = 1\\cfrac{\\bm{x^2}}{\\bm{169}}+\\cfrac{\\bm{y^2}} {\\bm{144}} \\bm{= 1}<\/p>\n D\u00e9terminer l’\u00e9quation de l’ellipse suivante et les coordonn\u00e9es de ses foyers : <\/p>\n Les sommets horizontaux de l’ellipse sont les points (-4,1) et (10,1). Par cons\u00e9quent, son diam\u00e8tre horizontal et son rayon sont : ” title=”Rendered by QuickLaTeX.com” height=”252″ width=”2047″ style=”vertical-align: -20px;”><\/p>\n d_h=10-(-4) =14 a =\\cfrac{14}{2} = 7<\/p>\n d_v=6-(-4) =10 b =\\cfrac{10}{2} = 5<\/p>\n C_x= \\cfrac{10+(-4)}{2} = \\cfrac{6}{2} =3 C_y= \\cfrac{6+(-4)}{2} = \\cfrac{2}{ 2} = 1C(3.1)<\/p>\n \\cfrac{(x-x_0)^2}{a^2}+\\cfrac{(y-y_0)^2}{b^2} = 1\\cfrac{(x-3)^2}{7^2 }+\\cfrac{(y-1)^2}{5^2} =1\\cfrac{\\bm{(x-3)^2}}{\\bm{49}}+\\cfrac{\\bm{( y-1)^2}}{\\bm{25}} \\bm{= 1}<\/p>\n a^2=b^2+c^2 c^2=a^2-b^2 c=\\sqrt{a^2-b^2} = \\sqrt{7^2-5^2}=\\sqrt {24}<\/p>\n \\sqrt{24}<\/p>\n C(3,1) \\bm{F\\esquerda(3+\\sqrt{24},1}\\direita)} \\bm{F\\esquerda(3-\\sqrt{24},1}\\direita)}<\/p>\n Calculez l’\u00e9quation de l’ellipse qui r\u00e9pond aux caract\u00e9ristiques suivantes :<\/p>\n On peut calculer la demi-focale \u00e0 partir de la focale : ” title=”Rendered by QuickLaTeX.com” height=”185″ width=”1667″ style=”vertical-align: -19px;”><\/p>\n 2c = 6 c=\\cfrac{6}{2} c=3<\/p>\n d(P,F) + d(P,F’)= 2a 3+5= 2a 8= 2a \\cfrac{8}{2}= a 4= a<\/p>\n a^2=b^2+c^2 b^2=a^2-c^2 b=\\sqrt{a^2-c^2} = \\sqrt{4^2-3^2}=\\sqrt {7}<\/p>\n \\cfrac{(x-x_0)^2}{a^2}+\\cfrac{(y-y_0)^2}{b^2} = 1\\cfrac{(x-0)^2}{4^2 }+\\cfrac{(y-0)^2}{\\left(\\sqrt{7}\\right)^2} =1\\cfrac{\\bm{x^2}}{\\bm{16}}+\\ cfrac{\\bm{y^2}}{\\bm{7}} \\bm{= 1}$<\/p>\n Por fim, se este artigo foi \u00fatil para voc\u00ea, certamente tamb\u00e9m se interessar\u00e1 por nossas p\u00e1ginas sobre a f\u00f3rmula da hip\u00e9rbole<\/a> e a f\u00f3rmula da par\u00e1bola<\/a> . Voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o detalhada do que s\u00e3o a hip\u00e9rbole e a par\u00e1bola, suas equa\u00e7\u00f5es, suas caracter\u00edsticas, exemplos, exerc\u00edcios resolvidos,\u2026<\/p>\n","protected":false},"excerpt":{"rendered":" Aqui voc\u00ea descobrir\u00e1 como a equa\u00e7\u00e3o (f\u00f3rmula) da elipse \u00e9 calculada, tendo ela a origem como centro ou n\u00e3o. Voc\u00ea tamb\u00e9m descobrir\u00e1 quais s\u00e3o os elementos da elipse, como calcul\u00e1-los e para que servem. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos de equa\u00e7\u00f5es de elipse. F\u00f3rmula da equa\u00e7\u00e3o da elipse A f\u00f3rmula para …<\/p>\n![\"\\cfrac{(x-x_0)^2}{a^2}+\\cfrac{(y-y_0)^2}{b^2}](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e29350c8a9f9271d7c58bb5636661eae_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n\n
![\"x_0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87f2a80bc63f8d7bc3df68c45a787402_l3.png\" "\\"Rendered")
<\/p>\n![\"y_0\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d37dc47669aa63f72480eae663d99287_l3.png\" "\\"Rendered")
<\/p>\n![\"C(x_0,y_0)\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a54160c9f13bae428a2471d905abd6f7_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n![\"a\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" "\\"Rendered")
<\/p>\n![\"b\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" "\\"Rendered")
<\/p>\n![\"f\u00f3rmula](\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-dune-ellipse.webp\")
<\/figure>\n<\/div>\n<\/span> Equa\u00e7\u00e3o da elipse centrada na origem<\/span><\/h3>\n
<\/p>\n<\/p>\n<\/p>\n<\/p>\n![\"\\cfrac{\\bm{x^2}}{\\bm{a^2}}+\\cfrac{\\bm{y^2}}{\\bm{b^2}}](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7821573c61c10361101554eb56041901_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> elementos da elipse<\/span><\/h2>\n
\n
![\"elementos](\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/elements-dellipse.webp\")
<\/figure>\n<\/div>\n<\/span> Rela\u00e7\u00e3o entre elementos de uma elipse<\/span><\/h3>\n
![\"d(P,F)](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7cef5996a2621318273bd54d01594941_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"d(P,F)\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6be809958050006a77cc59c5b7c32557_l3.png\" "\\"Rendered")
<\/p>\n![\"d(P,F')\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52cd58325f7f5f8ae50bf05b32b7ed55_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/p>\n![\"equa\u00e7\u00e3o](\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/relation-delements-dellipse.webp\")
<\/figure>\n<\/div>\n![\"a^2=b^2+c^2\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f07be3767557be2f8c17fc9a226a2506_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Excentricidade da elipse<\/span><\/h2>\n
![\"e](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01e68e598b53e74e9420afdb1bf6ab66_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"c\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n![\"excentricidade](\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/excentricite-dellipse.webp\")
<\/figure>\n<\/div>\n![\"0\n\n<h2](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7cba3912f2e788be4e73f1e18c9fb21_l3.png\")
<\/span> Exemple de calcul de l’\u00e9quation de l’ellipse<\/span><\/h2>\n\n
![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ac0d898ef827d924f8a7972d18a3d37_l3.png\" "\\"Rendered")
<\/p>\n![\"\n\n<h2](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c8518a77b08b28dd3989532a9c1a0bd_l3.png\")
<\/span> Probl\u00e8mes r\u00e9solus de l’\u00e9quation de l’ellipse<\/span><\/h2>\n Exercice 1<\/h3>\n
![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1a535f3c26d0c91d21ff2802c71cb131_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e674e398f6c61cceb56ebc7d6849b2b7_l3.png\")
\n!["\u00e9quation]("https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/centre-de-lellipse-de-lequation-a-lexterieur-de-lorigine.webp")
<\/figure>\n<\/div>\n Exercice 2<\/h3>\n
![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bcf056c99846b69f1bf4ed5ce1e6552a_l3.png\" "\\"Rendered")
<\/p>\n![\"\n\n<div](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e53a75087af9221fe85fa404a4045ff_l3.png\")
<\/div>\n Exercice 3<\/h3>\n
!["exercices]("https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-de-lequation-de-lellipse.webp")
<\/figure>\n<\/div>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76aa999f562c86113192c06e01991927_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ac099c741c43ec962a36c7b2bba5d06_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-51217c6f75e2ad233a651376f2ded0e0_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-310ed81b139fc6b5d3902b75bed66c9b_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1653b114c298901c587b9af56e4b0c40_l3.png\" "\\"Rendered")
<\/p>\n![\"unit\u00e9s](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f5155fae442053fd60dec7ee847fe0f_l3.png\" "\\"Rendered")
<\/p>\n![\"\n\n<div](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d494da27f7cde61e219586567d178c8_l3.png\")
<\/div>\n Exercice 4<\/h3>\n
\n
![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae016b9237aee40e4130230eb495fe6b_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-615d5b2ccb4a343777d9d707806526ab_l3.png\" "\\"Rendered")
<\/p>\n![\"](\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-98a1c514fb7344ec3be2558c5a559feb_l3.png\" "\\"Rendered")
<\/p>\n