{"id":28991,"date":"2021-05-05T13:00:10","date_gmt":"2021-05-05T13:00:10","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28991"},"modified":"2024-09-22T01:53:44","modified_gmt":"2024-09-22T01:53:44","slug":"equacao-das-hiperboles-e-sua-deducao","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/equacao-das-hiperboles-e-sua-deducao\/","title":{"rendered":"Equa\u00e7\u00e3o das Hip\u00e9rboles e sua Dedu\u00e7\u00e3o"},"content":{"rendered":"<p><center><\/p>\n<h1>Equa\u00e7\u00e3o das Hip\u00e9rboles e sua Dedu\u00e7\u00e3o<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\nNesta aula, exploraremos a defini\u00e7\u00e3o geom\u00e9trica da hip\u00e9rbole, contrastando-a com a elipse, e deduziremos sua equa\u00e7\u00e3o geral e can\u00f4nica.<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>Definir<\/strong> geometricamente o que \u00e9 uma hip\u00e9rbole.<\/li>\n<li><strong>Deducir<\/strong> a equa\u00e7\u00e3o geral e can\u00f4nica das hip\u00e9rboles a partir de sua defini\u00e7\u00e3o geom\u00e9trica.<\/li>\n<li><strong>Identificar<\/strong> as diferen\u00e7as entre as elipses e as hip\u00e9rboles em termos das dist\u00e2ncias focais.<\/li>\n<\/ol>\n<p>   <strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n<a href=\"#1\">Defini\u00e7\u00e3o Geom\u00e9trica de Hip\u00e9rbole<\/a><br \/>\n<a href=\"#2\">Dedu\u00e7\u00e3o da Equa\u00e7\u00e3o das Hip\u00e9rboles<\/a><br \/>\n<a href=\"#3\">Equa\u00e7\u00e3o Geral das Hip\u00e9rboles<\/a><br \/>\n<a href=\"#4\">Equa\u00e7\u00e3o Can\u00f4nica das Hip\u00e9rboles<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/1Aearz-E3bk\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a>   <\/p>\n<h2>Defini\u00e7\u00e3o Geom\u00e9trica de Hip\u00e9rbole<\/h2>\n<p style=\"text-align: justify;\">Anteriormente, revisamos a equa\u00e7\u00e3o das elipses e circunfer\u00eancias e descobrimos que t\u00eam a forma <span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx + cy^2 + dy + e = 0<\/span>, onde <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> s\u00e3o duas quantidades distintas de zero e com o mesmo sinal. Sobre isso comentamos que se <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> t\u00eam sinais opostos, ent\u00e3o em vez de uma elipse teremos uma Hip\u00e9rbole. Nada mais dissemos sobre essas curvas, e agora vamos preencher essa lacuna. Completaremos nosso estudo definindo o que \u00e9 geometricamente uma hip\u00e9rbole e, a partir disso, obteremos a equa\u00e7\u00e3o geral e can\u00f4nica das hip\u00e9rboles.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=176s\" target=\"_blank\" rel=\"noopener\"><strong>Por um lado, temos que a elipse \u00e9 definida<\/strong><\/a> como o conjunto de todos os pontos tais que a soma de suas dist\u00e2ncias em rela\u00e7\u00e3o a outros dois pontos que chamamos de focos \u00e9 sempre o mesmo valor. De forma semelhante, e em oposi\u00e7\u00e3o, a hip\u00e9rbole \u00e9 definida como a cole\u00e7\u00e3o de todos os pontos tais que o valor absoluto da diferen\u00e7a entre as dist\u00e2ncias dos pontos focais \u00e9 sempre o mesmo valor.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-BNQiwaq_OJs\/YJGda6VHOmI\/AAAAAAAAFDo\/edTQHWwLGGQwyGszR7c-7H74a09ASsK2gCLcBGAsYHQ\/s0\/hiperbola%2Bdefinici%25C3%25B3n%2Bgr%25C3%25A1fica.PNG\" alt=\"Defini\u00e7\u00e3o Geom\u00e9trica de Hip\u00e9rbole\" class=\" aligncenter lazyload\" width=\"493\" height=\"340\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-BNQiwaq_OJs\/YJGda6VHOmI\/AAAAAAAAFDo\/edTQHWwLGGQwyGszR7c-7H74a09ASsK2gCLcBGAsYHQ\/s0\/hiperbola%2Bdefinici%25C3%25B3n%2Bgr%25C3%25A1fica.PNG\" alt=\"Defini\u00e7\u00e3o Geom\u00e9trica de Hip\u00e9rbole\" class=\" aligncenter lazyload\" width=\"493\" height=\"340\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Ou seja, satisfaz a rela\u00e7\u00e3o<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">|d(f_1,P) - d(f_2,P)| = 2a<\/span>\n<p style=\"text-align: justify;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u00e9 qualquer n\u00famero real fixo.<\/p>\n<p style=\"text-align: justify;\">Isso gera, na verdade, duas equa\u00e7\u00f5es, a saber: <span class=\"katex-eq\" data-katex-display=\"false\">d(f_1,P) - d(f_2,P) = 2a<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">d(f_2,P) - d(f_1,P) = 2a<\/span>, uma para cada ramo da hip\u00e9rbole.<\/p>\n<p><a name=\"2\"><\/a>  <\/p>\n<h2>Dedu\u00e7\u00e3o da Equa\u00e7\u00e3o das Hip\u00e9rboles<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=331s\" target=\"_blank\" rel=\"noopener\"><strong>A partir da defini\u00e7\u00e3o geom\u00e9trica, \u00e9 poss\u00edvel obter<\/strong><\/a> a representa\u00e7\u00e3o alg\u00e9brica das hip\u00e9rboles. Para isso, come\u00e7aremos pelo caso mais simples, e a partir da\u00ed, faremos as generaliza\u00e7\u00f5es. Nosso racioc\u00ednio ser\u00e1 realizado para um \u00fanico ramo da hip\u00e9rbole; o racioc\u00ednio para o outro ramo \u00e9 completamente an\u00e1logo.<\/p>\n<h3>Dedu\u00e7\u00e3o da Forma Simplificada<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=356s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremos dois pontos focais<\/strong><\/a> <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> O ponto <span class=\"katex-eq\" data-katex-display=\"false\">p = (x,y)<\/span> estar\u00e1 contido na hip\u00e9rbole se<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-SMOUgyC1lM4\/YJGg_MIkJTI\/AAAAAAAAFDw\/6JzXOcfZi70lpvTZtbC6y26AvTQzcnWNgCLcBGAsYHQ\/s0\/hiperbola%2Bcentrada%2Ben%2Bel%2Borigen.PNG\" alt=\"Hip\u00e9rbole centrada na origem\" class=\"aligncenter  lazyload\" width=\"342\" height=\"288\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-SMOUgyC1lM4\/YJGg_MIkJTI\/AAAAAAAAFDw\/6JzXOcfZi70lpvTZtbC6y26AvTQzcnWNgCLcBGAsYHQ\/s0\/hiperbola%2Bcentrada%2Ben%2Bel%2Borigen.PNG\" alt=\"Hip\u00e9rbole centrada na origem\" class=\"aligncenter  lazyload\" width=\"342\" height=\"288\" \/><\/noscript><\/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>   &nbsp;<\/p>\n<p style=\"text-align: justify;\">E, a partir da\u00ed, segue-se o seguinte racioc\u00ednio:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\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>; equa\u00e7\u00e3o das hip\u00e9rboles<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x^2 + 2xc + c^2 + y^2} - \\sqrt{x^2 - 2xc + c^2 + y^2} = 2a<\/span><\/td>\n<td>; expandindo os quadrados<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x^2 + 2xc + c^2 + y^2} = 2a + \\sqrt{x^2 - 2xc + c^2 + y^2}<\/span><\/td>\n<td>; redistribuindo termos<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\color{red}{x^2} + 2xc + \\color{purple}{c^2} + \\color{violet}{y^2} = 4a^2 + 4a\\sqrt{x^2 - 2xc + c^2 + y^2} + \\color{red}{x^2} - 2xc + \\color{purple}{c^2} + \\color{violet}{y^2}<\/span><\/td>\n<td>; elevando ambos os lados ao quadrado<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 2xc = 4a^2 + 4a\\sqrt{x^2 - 2xc + c^2 + y^2} - 2xc <\/span><\/td>\n<td>; eliminando termos semelhantes<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 4xc = 4a^2 + 4a\\sqrt{x^2 - 2xc + c^2 + y^2} <\/span><\/td>\n<td>; redistribuindo termos semelhantes<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> xc = a^2 + a\\sqrt{x^2 - 2xc + c^2 + y^2} <\/span><\/td>\n<td>; Simplificando termos semelhantes<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> xc - a^2 = a\\sqrt{x^2 - 2xc + c^2 + y^2} <\/span><\/td>\n<td>; simplificando termos semelhantes<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2c^2 -2xca^2 + a^4 = a^2(x^2 - 2xc + c^2 + y^2) <\/span><\/td>\n<td>; elevando ambos os lados ao quadrado<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2c^2 \\color{red}{-2xca^2} + a^4 = a^2x^2 \\color{red}{- 2xca^2} + a^2c^2 + a^2y^2 <\/span><\/td>\n<td>; operando par\u00eanteses<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2c^2 + a^4 = a^2x^2 + a^2c^2 + a^2y^2 <\/span><\/td>\n<td>; eliminando termos semelhantes<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2(c^2 - a^2) - a^2y^2 = a^2c^2 - a^4 = a^2(c^2 - a^2) <\/span><\/td>\n<td>; reagrupando termos<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{x^2}{a^2} - \\frac{y^2}{c^2 - a^2} = 1 <\/span><\/td>\n<td>; reagrupando termos<\/td>\n<\/tr>\n<tr><\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Para esta \u00faltima express\u00e3o, assim como para as elipses, tomamos <span class=\"katex-eq\" data-katex-display=\"false\">b^2=c^2-a^2<\/span> e chegamos \u00e0 equa\u00e7\u00e3o das elipses:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{ \\left(\\frac{x}{a}\\right)^2 - \\left(\\frac{y}{b}\\right)^2 = 1 }<\/span>\n<p><a name=\"3\"><\/a>     <\/p>\n<h2>Equa\u00e7\u00e3o Geral das Hip\u00e9rboles<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong>Para obter a equa\u00e7\u00e3o geral<\/strong><\/a> das hip\u00e9rboles, basta tomar a que acabamos de obter e aplicar as transforma\u00e7\u00f5es de posi\u00e7\u00e3o:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto x-h<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto y-k<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">e com isso obtemos automaticamente a equa\u00e7\u00e3o geral das elipses com centro <span class=\"katex-eq\" data-katex-display=\"false\">(h,k)<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{ \\left(\\frac{x-h}{a}\\right)^2 - \\left(\\frac{y-k}{b}\\right)^2 = 1 }<\/span>\n<p><a name=\"4\"><\/a>     <\/p>\n<h2>Equa\u00e7\u00e3o Can\u00f4nica das Hip\u00e9rboles<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=974s\" target=\"_blank\" rel=\"noopener\"><strong>E se agora tomarmos a equa\u00e7\u00e3o geral<\/strong><\/a> das elipses e a desenvolvemos, chegaremos \u00e0 express\u00e3o can\u00f4nica:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{x-h}{a}\\right)^2 - \\left(\\frac{y-k}{b}\\right)^2 = 1<\/span><\/td>\n<td>; Equa\u00e7\u00e3o geral das hip\u00e9rboles<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 (x^2 - 2xh + h^2) - a^2(y^2-2ky + y^2) = a^2b^2<\/span><\/td>\n<td>; resolvendo os quadrados e multiplicando tudo por <span class=\"katex-eq\" data-katex-display=\"false\">a^2b^2<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2 - 2hb^2x + h^2b^2 - a^2 y^2+ 2k a^2 y - a^2 k^2 = a^2b^2<\/span><\/td>\n<td>; resolvendo par\u00eanteses<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2 - (2hb^2) x - a^2 y^2+ (2k a^2) y - (a^2b^2 + a^2 k^2 - h^2b^2) = 0 <\/span><\/td>\n<td>; Agrupando termos semelhantes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Esta \u00faltima express\u00e3o \u00e9 da forma <span class=\"katex-eq\" data-katex-display=\"false\">Ax^2+Bx + Cy^2 + Dy + E = 0,<\/span> onde <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> s\u00e3o sempre diferentes de zero e com sinais opostos, como j\u00e1 hav\u00edamos antecipado quando estud\u00e1vamos as elipses.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equa\u00e7\u00e3o das Hip\u00e9rboles e sua Dedu\u00e7\u00e3o Resumo: Nesta aula, exploraremos a defini\u00e7\u00e3o geom\u00e9trica da hip\u00e9rbole, contrastando-a com a elipse, e deduziremos sua equa\u00e7\u00e3o geral e can\u00f4nica. Objetivos de Aprendizagem: Ao final desta aula, o aluno ser\u00e1 capaz de: Definir geometricamente o que \u00e9 uma hip\u00e9rbole. Deducir a equa\u00e7\u00e3o geral e can\u00f4nica das hip\u00e9rboles a partir [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28988,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":46,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-28991","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Equa\u00e7\u00e3o das Hip\u00e9rboles e sua Dedu\u00e7\u00e3o - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda tudo sobre as hip\u00e9rboles: defini\u00e7\u00e3o geom\u00e9trica, dedu\u00e7\u00e3o da sua equa\u00e7\u00e3o geral e can\u00f4nica, diferen\u00e7as com as elipses, e como resolver equa\u00e7\u00f5es passo a passo. 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