{"id":28962,"date":"2024-09-21T13:28:21","date_gmt":"2024-09-21T13:28:21","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28962"},"modified":"2024-09-21T13:28:21","modified_gmt":"2024-09-21T13:28:21","slug":"equacao-das-elipses-e-circunferencias","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/equacao-das-elipses-e-circunferencias\/","title":{"rendered":"Equa\u00e7\u00e3o das Elipses e Circunfer\u00eancias"},"content":{"rendered":"<p><center><\/p>\n<h1>Equa\u00e7\u00e3o das Elipses e Circunfer\u00eancias<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\n   Esta aula explica a obten\u00e7\u00e3o da equa\u00e7\u00e3o das elipses a partir de sua defini\u00e7\u00e3o geom\u00e9trica, que estabelece que a soma das dist\u00e2ncias de qualquer ponto da elipse a dois focos fixos \u00e9 constante. Atrav\u00e9s de um desenvolvimento alg\u00e9brico detalhado, \u00e9 deduzida a equa\u00e7\u00e3o geral das elipses e sua forma can\u00f4nica, bem como a conex\u00e3o entre as elipses e as circunfer\u00eancias, mostrando que uma circunfer\u00eancia \u00e9 um caso especial de elipse quando os semi-eixos s\u00e3o iguais.<br \/>\n   <\/em><\/p>\n<p>   <strong>Objetivos de Aprendizagem:<\/strong><br \/>\n   Ao final desta aula, o aluno ser\u00e1 capaz de:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Deducir<\/strong> a equa\u00e7\u00e3o das elipses a partir de sua defini\u00e7\u00e3o geom\u00e9trica.<\/li>\n<li><strong>Reconhecer<\/strong> tanto a forma geral quanto a forma can\u00f4nica da equa\u00e7\u00e3o das elipses.<\/li>\n<\/ol>\n<p>   \u00cdNDICE DE CONTE\u00daDOS<br \/>\n   <a href=\"#1\">Formula\u00e7\u00e3o geom\u00e9trica<\/a><br \/>\n   <a href=\"#2\">Obten\u00e7\u00e3o da equa\u00e7\u00e3o das elipses<\/a><br \/>\n   <a href=\"#3\">Equa\u00e7\u00e3o geral das elipses<\/a><br \/>\n   <a href=\"#4\">Equa\u00e7\u00e3o can\u00f4nica das elipses<\/a><br \/>\n   <a href=\"#5\">Redu\u00e7\u00e3o \u00e0 Equa\u00e7\u00e3o das Circunfer\u00eancias<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HHiC0bp-Vyc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n   <a name=\"1\"><\/a><\/p>\n<h2>Formula\u00e7\u00e3o geom\u00e9trica<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=133s\" target=\"_blank\" rel=\"noopener\"><strong>Para obter a equa\u00e7\u00e3o que descreve as elipses,<\/strong> <\/a> devemos raciocinar, assim como com as par\u00e1bolas, sobre o significado geom\u00e9trico delas. Uma elipse \u00e9 o conjunto de todos os pontos do plano tal que a soma das dist\u00e2ncias entre eles e dois pontos chamados de focos \u00e9 sempre a mesma.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-rHroj77w4-o\/YIhoGfTvE_I\/AAAAAAAAFAw\/2Yoa3Q2yrmknQMPObDz8wuyDoOehCug5QCLcBGAsYHQ\/s0\/elipse.PNG\" alt=\"Elipse\" class=\" aligncenter lazyload\" width=\"338\" height=\"241\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-rHroj77w4-o\/YIhoGfTvE_I\/AAAAAAAAFAw\/2Yoa3Q2yrmknQMPObDz8wuyDoOehCug5QCLcBGAsYHQ\/s0\/elipse.PNG\" alt=\"Elipse\" class=\" aligncenter lazyload\" width=\"338\" height=\"241\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Ou seja, se cumprir\u00e1 que:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">d(f_1,p) + d(f_2,p) = constante<\/span>\n<p>   <a name=\"2\"><\/a><\/p>\n<h2>Obten\u00e7\u00e3o da equa\u00e7\u00e3o das elipses<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=311s\" target=\"_blank\" rel=\"noopener\"><strong>A partir da defini\u00e7\u00e3o geom\u00e9trica<\/strong><\/a> das elipses podemos obter uma express\u00e3o alg\u00e9brica que as descreve. Para fazer isso com facilidade, no entanto, recorreremos a algumas simplifica\u00e7\u00f5es. Consideraremos, sem perda de generalidade, que os focos t\u00eam posi\u00e7\u00f5es <span class=\"katex-eq\" data-katex-display=\"false\">f_1 =(-c,0)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">f_2 =(c,0),<\/span> deste modo, se um ponto qualquer <span class=\"katex-eq\" data-katex-display=\"false\">p=(x,y)<\/span> fizer parte da elipse, ent\u00e3o ser\u00e1 cumprido que:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a<\/span>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-LtAamnh5D78\/YIiBshjM70I\/AAAAAAAAFA4\/hGiHx6jf_nMOOUHfH-Ywj34TyDJDGEv-wCLcBGAsYHQ\/s0\/ecuacion%2Bde%2Blas%2Belipses.PNG\" alt=\"Equa\u00e7\u00e3o das Elipses\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-LtAamnh5D78\/YIiBshjM70I\/AAAAAAAAFA4\/hGiHx6jf_nMOOUHfH-Ywj34TyDJDGEv-wCLcBGAsYHQ\/s0\/ecuacion%2Bde%2Blas%2Belipses.PNG\" alt=\"Equa\u00e7\u00e3o das Elipses\" class=\" aligncenter lazyload\" width=\"412\" height=\"333\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span> \u00e9 uma constante fixa. A partir disso, podemos construir o seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x+c)^2 + y^2} + \\sqrt{(x-c)^2 + y^2} = 2a<\/span><\/td>\n<td>; Defini\u00e7\u00e3o geom\u00e9trica da elipse<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x-c)^2 + y^2} = 2a - \\sqrt{(x+c)^2 + y^2}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(x-c)^2 + \\cancel{y^2} = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + (x+c)^2 + \\cancel{y^2}<\/span><\/td>\n<td>; elevando ao quadrado (1)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(x-c)^2 = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + (x+c)^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\cancel{x^2} -2xc + \\cancel{c^2} = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} + \\cancel{x^2} +2xc + \\cancel{c^2} <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">-2xc = 4a^2 -4a \\sqrt{(x+c)^2 + y^2} +2xc <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">4a \\sqrt{(x+c)^2 + y^2} = 4a^2 +4xc = 4(a^2 + xc) <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a \\sqrt{(x+c)^2 + y^2} = a^2 + xc <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a^2 [(x+c)^2 + y^2] = (a^2 + xc)^2 <\/span><\/td>\n<td>; elevando ao quadrado (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">a^2 [x^2 + 2xc + c2 + y^2] = a^4 +2a^2xc + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> a^2 x^2 + \\cancel{2xca^2} + a^2 c2 + a^2 y^2 = a^4 + \\cancel{2a^2xc} + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> a^2 x^2 + a^2 c2 + a^2 y^2 = a^4 + x^2c^2 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2 (a^2 - c^2) + a^2 y^2 = a^4 - a^2 c^2 =a^2(a^2-c^2) <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{x^2}{a^2} +\\dfrac{ y^2}{a^2-c^2} = 1 <\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt a^2 - c^2 =: b^2 <\/span><\/td>\n<td>; O n\u00famero representado por <span class=\"katex-eq\" data-katex-display=\"false\">b^2<\/span> \u00e9 positivo, como se v\u00ea na figura.<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{\\dfrac{x^2}{a^2} +\\dfrac{ y^2}{b^2} = 1}<\/span><\/td>\n<td>; De (3) e (4)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\left(\\dfrac{x}{a}\\right)^2 + \\left(\\dfrac{y}{b}\\right)^2 = 1}<\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Este \u00e9 o que chamamos de \u00abequa\u00e7\u00e3o das elipses\u00bb.<\/p>\n<p>   <a name=\"3\"><\/a><\/p>\n<h2>Equa\u00e7\u00e3o geral das elipses<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=706s\" target=\"_blank\" rel=\"noopener\"><strong>A equa\u00e7\u00e3o que acabamos de obter<\/strong><\/a> pode ser levada \u00e0 sua forma geral por meio de transforma\u00e7\u00f5es de transla\u00e7\u00e3o, substituindo <span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto (x-h)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto (y-k).<\/span> Com isso, chegamos \u00e0 forma geral da equa\u00e7\u00e3o das elipses:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\left(\\dfrac{x-h}{a}\\right)^2 + \\left(\\dfrac{y-k}{b}\\right)^2 = 1}<\/span>\n<p style=\"text-align: justify;\">Esta \u00e9 uma elipse com centro no ponto <span class=\"katex-eq\" data-katex-display=\"false\">(h,k)<\/span>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-lkxt91FvMTs\/YIiQaL9wpII\/AAAAAAAAFBA\/sUxc6ajd6tcPymC8g4oh3M0l2CTI-xOvgCLcBGAsYHQ\/s0\/elipsegeneral.PNG\" alt=\"elipse geral\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-lkxt91FvMTs\/YIiQaL9wpII\/AAAAAAAAFBA\/sUxc6ajd6tcPymC8g4oh3M0l2CTI-xOvgCLcBGAsYHQ\/s0\/elipsegeneral.PNG\" alt=\"elipse geral\" class=\" aligncenter lazyload\" width=\"469\" height=\"373\" \/><\/noscript><br \/>\n   <a name=\"4\"><\/a><\/p>\n<h2>Equa\u00e7\u00e3o can\u00f4nica das elipses<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=761s\" target=\"_blank\" rel=\"noopener\"><strong>Fazendo \u00e1lgebra sobre isso<\/strong><\/a> chegamos \u00e0 equa\u00e7\u00e3o can\u00f4nica das elipses:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td width=\"50\">(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\dfrac{x-h}{a}\\right)^2 + \\left(\\dfrac{y-k}{b}\\right)^2 = 1<\/span><\/td>\n<td>; equa\u00e7\u00e3o geral das elipses<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 (x-h)^2 + a^2(y-k)^2 = a^2 b^2<\/span><\/td>\n<td>; Multiplicar tudo por <span class=\"katex-eq\" data-katex-display=\"false\">a^2b^2<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 [x^2-2xh+h^2] + a^2[y^2-2yk + k^2] = a^2 b^2<\/span><\/td>\n<td>; desenvolver quadrados<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2-2hb^2 x + h^2b^2 + a^2 y^2-2ka^2y + k^2a^2 = a^2 b^2<\/span><\/td>\n<td>; desenvolver par\u00eanteses<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2- 2hb^2 x + a^2 y^2-2ka^2y +(h^2b^2 + k^2a^2 - a^2 b^2) = 0 <\/span><\/td>\n<td>; agrupar termos constantes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Sobre esta \u00faltima express\u00e3o, podemos fazer as substitui\u00e7\u00f5es <span class=\"katex-eq\" data-katex-display=\"false\">A:=b^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">B:=-2hb^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">C:=a^2,<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">D:=-2ka^2<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">E:=h^2b^2 + k^2a^2 - a^2 b^2.<\/span> Assim, veremos que as elipses ser\u00e3o descritas por equa\u00e7\u00f5es da forma:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Ax^2 + Bx + Cy^2 + Dy + E = 0<\/span>\n<p style=\"text-align: justify;\">Isto \u00e9 o que chamamos de \u00abEqua\u00e7\u00e3o Can\u00f4nica das Elipses\u00bb.<\/p>\n<p style=\"text-align: justify;\">Desses desenvolvimentos, podem ser extra\u00eddas algumas restri\u00e7\u00f5es sobre as constantes da equa\u00e7\u00e3o can\u00f4nica. A mais importante \u00e9 que <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> devem ter o mesmo sinal; caso contr\u00e1rio, j\u00e1 n\u00e3o estaremos falando de uma elipse, mas sim de uma hip\u00e9rbole. Existem mais restri\u00e7\u00f5es sobre as constantes da representa\u00e7\u00e3o can\u00f4nica, mas falar delas agora n\u00e3o \u00e9 o mais eficiente; veremos em detalhe quando revisarmos a caracteriza\u00e7\u00e3o das elipses e hip\u00e9rboles.<\/p>\n<p>   <a name=\"5\"><\/a><\/p>\n<h2>Redu\u00e7\u00e3o \u00e0 Equa\u00e7\u00e3o das Circunfer\u00eancias<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=HHiC0bp-Vyc&amp;t=948s\" target=\"_blank\" rel=\"noopener\"><strong>Uma coisa que revisaremos quando falarmos<\/strong><\/a> sobre a caracteriza\u00e7\u00e3o das elipses \u00e9 que as constantes <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> da equa\u00e7\u00e3o geral correspondem aos semi-eixos da elipse. Se tomarmos ambos os semi-eixos e os igualarmos, fazendo <span class=\"katex-eq\" data-katex-display=\"false\">a=b=r,<\/span> ent\u00e3o a elipse se transformar\u00e1 em uma circunfer\u00eancia de raio <span class=\"katex-eq\" data-katex-display=\"false\">r.<\/span>\n<h3>Equa\u00e7\u00e3o geral das circunfer\u00eancias<\/h3>\n<p style=\"text-align: justify;\">Dessa forma, obt\u00e9m-se a equa\u00e7\u00e3o geral das circunfer\u00eancias como:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">(x-h)^2 + (y-k)^2 = r^2<\/span>\n<h3>Equa\u00e7\u00e3o can\u00f4nica das circunfer\u00eancias<\/h3>\n<p style=\"text-align: justify;\">De forma an\u00e1loga, obt\u00e9m-se a equa\u00e7\u00e3o can\u00f4nica das circunfer\u00eancias:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Ax^2 + Bx + Cy^2 + Dy + E = 0<\/span>\n<p style=\"text-align: justify;\">Em sua forma can\u00f4nica coincide com as elipses, pois as circunfer\u00eancias, como vimos, s\u00e3o um caso particular de elipse.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equa\u00e7\u00e3o das Elipses e Circunfer\u00eancias Resumo: Esta aula explica a obten\u00e7\u00e3o da equa\u00e7\u00e3o das elipses a partir de sua defini\u00e7\u00e3o geom\u00e9trica, que estabelece que a soma das dist\u00e2ncias de qualquer ponto da elipse a dois focos fixos \u00e9 constante. Atrav\u00e9s de um desenvolvimento alg\u00e9brico detalhado, \u00e9 deduzida a equa\u00e7\u00e3o geral das elipses e sua forma [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28959,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":7,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-28962","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-e-geometria","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Equa\u00e7\u00e3o das Elipses e Circunfer\u00eancias - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda a deduzir a equa\u00e7\u00e3o das elipses e sua rela\u00e7\u00e3o com as circunfer\u00eancias. 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