{"id":29023,"date":"2021-05-05T13:00:44","date_gmt":"2021-05-05T13:00:44","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29023"},"modified":"2024-09-22T04:31:39","modified_gmt":"2024-09-22T04:31:39","slug":"secoes-conicas-caracterizacao-e-grafico-de-parabolas-elipses-e-hiperboles","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/secoes-conicas-caracterizacao-e-grafico-de-parabolas-elipses-e-hiperboles\/","title":{"rendered":"Se\u00e7\u00f5es C\u00f4nicas: Caracteriza\u00e7\u00e3o e Gr\u00e1fico de Par\u00e1bolas, Elipses e Hip\u00e9rboles"},"content":{"rendered":"<p><center><\/p>\n<h1>Se\u00e7\u00f5es C\u00f4nicas: Caracteriza\u00e7\u00e3o e Gr\u00e1fico de Par\u00e1bolas, Elipses e Hip\u00e9rboles<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\nNesta aula, revisaremos as se\u00e7\u00f5es c\u00f4nicas (par\u00e1bolas, elipses e hip\u00e9rboles), come\u00e7ando por suas equa\u00e7\u00f5es can\u00f4nicas e gerais. Explicamos como identificar e caracterizar cada curva, focando em elementos-chave como o v\u00e9rtice, foco e eixo de simetria nas par\u00e1bolas, e a distin\u00e7\u00e3o entre elipses e hip\u00e9rboles de acordo com os sinais de seus coeficientes.<br \/>\n<\/em><br \/>\n<strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Reconhecer<\/strong> as equa\u00e7\u00f5es can\u00f4nicas das se\u00e7\u00f5es c\u00f4nicas (par\u00e1bolas, elipses, hip\u00e9rboles)<\/li>\n<li><strong>Calcular<\/strong> cada uma das caracter\u00edsticas das se\u00e7\u00f5es c\u00f4nicas: comprimento dos semi-eixos, dist\u00e2ncia focal, diretriz, etc.<\/li>\n<\/ol>\n<p><strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n<a href=\"#1\">Se\u00e7\u00f5es c\u00f4nicas<\/a><br \/>\n<a href=\"#2\">Revis\u00e3o das Par\u00e1bolas<\/a><br \/>\n<a href=\"#3\">Revis\u00e3o das Elipses e Hip\u00e9rboles<\/a><br \/>\n<a href=\"#4\">Caracteriza\u00e7\u00e3o da elipse<\/a><br \/>\n<a href=\"#5\">Caracteriza\u00e7\u00e3o da hip\u00e9rbole<\/a><br \/>\n<a href=\"#6\">Exerc\u00edcios Resolvidos<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/d21_9EHUv_M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Se\u00e7\u00f5es c\u00f4nicas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=126s\" target=\"_blank\" rel=\"noopener\"><strong>Chamamos de se\u00e7\u00f5es c\u00f4nicas<\/strong><\/a> todas as curvas que resultam da interse\u00e7\u00e3o da superf\u00edcie de um cone com um plano. A fam\u00edlia das se\u00e7\u00f5es c\u00f4nicas \u00e9 composta por circunfer\u00eancias, elipses e hip\u00e9rboles, todas curvas que j\u00e1 estudamos.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-5eckvpNxzlg\/YJJimcxkMYI\/AAAAAAAAFEA\/dfGTvXblcD4dZXSjpWvonYFN8O0EMNqtwCLcBGAsYHQ\/s0\/curvas-conicas-secciones-cono.png\" alt=\"Se\u00e7\u00f5es C\u00f4nicas\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-5eckvpNxzlg\/YJJimcxkMYI\/AAAAAAAAFEA\/dfGTvXblcD4dZXSjpWvonYFN8O0EMNqtwCLcBGAsYHQ\/s0\/curvas-conicas-secciones-cono.png\" alt=\"Se\u00e7\u00f5es C\u00f4nicas\" class=\" aligncenter lazyload\" width=\"531\" height=\"272\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Agora faremos uma revis\u00e3o das t\u00e9cnicas para reconhecer e caracterizar cada uma dessas curvas. Vamos nos concentrar especialmente nas formas can\u00f4nicas, pois s\u00e3o as mais frequentes e as que menos revelam informa\u00e7\u00f5es de forma expl\u00edcita. As equa\u00e7\u00f5es gerais, por outro lado, revelam quase toda a caracteriza\u00e7\u00e3o geom\u00e9trica.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Revis\u00e3o das Par\u00e1bolas<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=160s\" target=\"_blank\" rel=\"noopener\"><strong>Toda par\u00e1bola \u00e9 representada<\/strong><\/a> por uma equa\u00e7\u00e3o da forma<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y=ax^2 + bx + c,<\/span> com <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>\n<p style=\"text-align: justify;\">Em termos disso, obtivemos:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordenadas do V\u00e9rtice: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (x_0, y_0)=\\left( -\\dfrac{b}{2a}, c - \\dfrac{b^2}{4a} \\right)<\/span><\/li>\n<li>Posi\u00e7\u00e3o Focal: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f=\\dfrac{1}{4a}<\/span><\/li>\n<li>Coordenadas do Foco: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle foco=\\left( -\\dfrac{b}{2a}, c - \\dfrac{b^2}{4a} + f \\right) =\\left( -\\dfrac{b}{2a}, c + \\dfrac{1- b^2}{4a} \\right)<\/span><\/li>\n<li>Equa\u00e7\u00e3o da Diretriz: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle y= c - \\dfrac{b^2}{4a} - f = c - \\dfrac{1+b^2}{4a}<\/span><\/li>\n<li>Equa\u00e7\u00e3o do Eixo de Simetria: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x= -\\dfrac{b}{2a} <\/span><\/li>\n<li>Interse\u00e7\u00f5es com o eixo x (se existirem): <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x_{1,2}= \\dfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} <\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Com isso, j\u00e1 temos todas as informa\u00e7\u00f5es necess\u00e1rias para tra\u00e7ar o gr\u00e1fico de qualquer par\u00e1bola.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Revis\u00e3o das Elipses e Hip\u00e9rboles<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=449s\" target=\"_blank\" rel=\"noopener\"><strong>As elipses e hip\u00e9rboles, como vimos,<\/strong><\/a> t\u00eam uma express\u00e3o can\u00f4nica 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;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> s\u00e3o constantes diferentes de zero, e a partir do que estudamos, temos que:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Se <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> tiverem o mesmo sinal, \u00e9 uma elipse.<\/li>\n<li>Se <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> tiverem sinais opostos, \u00e9 uma hip\u00e9rbole.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Para separar claramente os dois casos, escreveremos que:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x + \\gamma y^2 + \\delta y + \\epsilon = 0<\/span> \u00e9 uma elipse.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha x^2+ \\beta x - \\gamma y^2 + \\delta y + \\epsilon = 0<\/span> \u00e9 uma hip\u00e9rbole.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Sendo <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta, \\gamma, \\delta<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> n\u00fameros reais quaisquer e <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span> sempre positivos. Escrever desta forma nos permite separar claramente os dois casos. A partir disso, podemos fazer as seguintes infer\u00eancias:<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Caracteriza\u00e7\u00e3o da Elipse<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=552s\" target=\"_blank\" rel=\"noopener\"><strong>Partindo da equa\u00e7\u00e3o can\u00f4nica,<\/strong><\/a> temos a seguinte dedu\u00e7\u00e3o:<\/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\">\\alpha x^2+ \\beta x + \\gamma y^2 + \\delta y + \\epsilon = 0<\/span><\/td>\n<td>; equa\u00e7\u00e3o can\u00f4nica das elipses.<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x^2+ \\dfrac{\\beta}{\\alpha }x\\right) + \\gamma \\left(y^2 + \\dfrac{\\delta}{\\gamma }y\\right) =- \\epsilon<\/span><\/td>\n<td>; fatorando e reagrupando termos<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2 + \\gamma \\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2 =\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<td>; completando quadrados e reagrupando termos<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} + \\gamma \\dfrac{\\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; dividindo tudo por <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\dfrac{1}{\\alpha }\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} + \\dfrac{\\left(y + \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\dfrac{1}{ \\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; rearranjando <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{ x - \\left(-\\dfrac{\\beta}{2 \\alpha }\\right)}{\\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 + \\left( \\dfrac{y - \\left(-\\dfrac{\\delta}{2 \\gamma } \\right)}{\\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 = 1<\/span><\/td>\n<td>; reestruturando com ra\u00edzes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">No desenvolvimento desta dedu\u00e7\u00e3o, o passo (3) \u00e9 particularmente delicado, pois se o coeficiente <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span> for negativo, ent\u00e3o a elipse n\u00e3o pode existir.<\/p>\n<p style=\"text-align: justify;\">Lembre-se de que a equa\u00e7\u00e3o geral das elipses \u00e9 da forma<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{x-h}{a} \\right)^2 + \\left(\\dfrac{y-k}{b} \\right)^2 = 1<\/span>\n<p style=\"text-align: justify;\">Com este \u00faltimo resultado, agora temos uma rela\u00e7\u00e3o direta entre os par\u00e2metros da f\u00f3rmula geral que nos permite revelar todas as informa\u00e7\u00f5es escondidas na express\u00e3o can\u00f4nica:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordenadas do centro: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (h,k) = \\left( -\\dfrac{\\beta}{2\\alpha}, -\\dfrac{\\delta}{2\\gamma}\\right)<\/span><\/li>\n<li>Comprimento do semi-eixo horizontal: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle a = \\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<li>Comprimento do semi-eixo vertical: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b = \\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } + \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Com isso, j\u00e1 \u00e9 poss\u00edvel reconhecer e tra\u00e7ar uma elipse diretamente a partir de sua forma can\u00f4nica. Seu gr\u00e1fico ser\u00e1 o seguinte:<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-aVD7YQ7DfL0\/YJKBv9QXeTI\/AAAAAAAAFEQ\/urCuFtrn-YYBQ_fVSGXsmhMqExFumag-ACLcBGAsYHQ\/s0\/elipse.PNG\" class=\" aligncenter lazyload\" width=\"414\" height=\"291\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-aVD7YQ7DfL0\/YJKBv9QXeTI\/AAAAAAAAFEQ\/urCuFtrn-YYBQ_fVSGXsmhMqExFumag-ACLcBGAsYHQ\/s0\/elipse.PNG\" class=\" aligncenter lazyload\" width=\"414\" height=\"291\" \/><\/noscript><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Caracteriza\u00e7\u00e3o da Hip\u00e9rbole<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=d21_9EHUv_M&amp;t=911s\" target=\"_blank\" rel=\"noopener\"><strong>Raciocinando de maneira completamente an\u00e1loga,<\/strong><\/a> voc\u00ea pode, a partir da equa\u00e7\u00e3o can\u00f4nica, fazer a caracteriza\u00e7\u00e3o completa das hip\u00e9rboles. Na verdade, a an\u00e1lise \u00e9 t\u00e3o an\u00e1loga que vou copiar e colar a an\u00e1lise das elipses e apenas modificar algumas partes.<\/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\">\\alpha x^2+ \\beta x - \\gamma y^2 + \\delta y + \\epsilon = 0<\/span><\/td>\n<td>; equa\u00e7\u00e3o can\u00f4nica das hip\u00e9rboles.<\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x^2+ \\dfrac{\\beta}{\\alpha }x\\right) - \\gamma \\left(y^2 - \\dfrac{\\delta}{\\gamma }y\\right) =- \\epsilon<\/span><\/td>\n<td>; fatorando e reagrupando termos<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2 - \\gamma \\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2 =\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<td>; completando quadrados e reagrupando termos<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\alpha \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} - \\gamma \\dfrac{\\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; dividindo tudo por <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon<\/span><\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\left( x + \\dfrac{\\beta}{2 \\alpha }\\right)^2}{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} - \\dfrac{\\left(y - \\dfrac{\\delta}{2 \\gamma } \\right)^2}{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)} = 1<\/span><\/td>\n<td>; rearranjando os termos <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\gamma<\/span><\/td>\n<\/tr>\n<tr>\n<td>(6)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{ x - \\left(-\\dfrac{\\beta}{2 \\alpha }\\right)}{\\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 - \\left( \\dfrac{y - \\left(\\dfrac{\\delta}{2 \\gamma } \\right)}{\\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}}\\right)^2 = 1<\/span><\/td>\n<td>; reestruturando com ra\u00edzes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">A partir disso, agora temos uma rela\u00e7\u00e3o direta entre a equa\u00e7\u00e3o can\u00f4nica e a equa\u00e7\u00e3o das hip\u00e9rboles que nos permitir\u00e1 confeccionar rapidamente seu gr\u00e1fico.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\dfrac{x-h}{a} \\right)^2 - \\left(\\dfrac{y-k}{b} \\right)^2 =1 <\/span>\n<p style=\"text-align: justify;\">Agora, ao contr\u00e1rio do que se faz com as elipses, aqui \u00e9 mais correto falar de \u00abcaixa geradora\u00bb como veremos na figura a seguir:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Coordenadas do centro: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (h,k) = \\left( -\\dfrac{\\beta}{2\\alpha}, \\dfrac{\\delta}{2\\gamma}\\right)<\/span><\/li>\n<li>Comprimento do semi-eixo horizontal: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle a = \\sqrt{\\dfrac{1}{\\alpha}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<li>Comprimento do semi-eixo vertical: <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b = \\sqrt{\\dfrac{1}{\\gamma}\\left(\\dfrac{\\beta^2}{4\\alpha } - \\dfrac{\\delta^2}{4\\gamma } - \\epsilon\\right)}<\/span><\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-bd0n_BuEFiE\/YJKJ1fPDhMI\/AAAAAAAAFEY\/-QjR2QbycSkKJihjHnwmdIDESYgNDyuBgCLcBGAsYHQ\/s0\/hiperbola.PNG\" alt=\"Hip\u00e9rbole\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-bd0n_BuEFiE\/YJKJ1fPDhMI\/AAAAAAAAFEY\/-QjR2QbycSkKJihjHnwmdIDESYgNDyuBgCLcBGAsYHQ\/s0\/hiperbola.PNG\" alt=\"Hip\u00e9rbole\" class=\" aligncenter lazyload\" width=\"428\" height=\"305\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Com os resultados dessas an\u00e1lises, j\u00e1 podemos tra\u00e7ar o gr\u00e1fico de qualquer membro da fam\u00edlia das se\u00e7\u00f5es c\u00f4nicas sem nenhuma dificuldade especial.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exerc\u00edcios Resolvidos<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/PKjQrcC0HG4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Se\u00e7\u00f5es C\u00f4nicas: Caracteriza\u00e7\u00e3o e Gr\u00e1fico de Par\u00e1bolas, Elipses e Hip\u00e9rboles Resumo: Nesta aula, revisaremos as se\u00e7\u00f5es c\u00f4nicas (par\u00e1bolas, elipses e hip\u00e9rboles), come\u00e7ando por suas equa\u00e7\u00f5es can\u00f4nicas e gerais. Explicamos como identificar e caracterizar cada curva, focando em elementos-chave como o v\u00e9rtice, foco e eixo de simetria nas par\u00e1bolas, e a distin\u00e7\u00e3o entre elipses e hip\u00e9rboles [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29020,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":20,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-29023","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>Se\u00e7\u00f5es C\u00f4nicas: Caracteriza\u00e7\u00e3o e Gr\u00e1fico de Par\u00e1bolas, Elipses e Hip\u00e9rboles - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"As se\u00e7\u00f5es c\u00f4nicas s\u00e3o as curvas 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Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. 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