{"id":17650,"date":"2020-08-27T15:01:05","date_gmt":"2020-08-27T18:01:05","guid":{"rendered":"https:\/\/beduka.com\/blog\/?p=17650"},"modified":"2020-08-27T15:01:06","modified_gmt":"2020-08-27T18:01:06","slug":"exercicios-sobre-hiperbole","status":"publish","type":"post","link":"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-sobre-hiperbole\/","title":{"rendered":"Exerc\u00edcios sobre Hip\u00e9rbole com Gabarito"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote\"><p>Hip\u00e9rboles adv\u00e9m de um corte que \u00e9 feito no cone (se\u00e7\u00e3o c\u00f4nica) onde formam-se duas \u201cmeia luas\u201d opostas. Hip\u00e9rbole \u00e9 conjunto de todos os pontos coplanares para os quais a diferen\u00e7a das dist\u00e2ncias a dois pontos fixos (focos) \u00e9 constante. Leia nosso resumo e fa\u00e7a nossa lista de exerc\u00edcios sobre Hip\u00e9rbole dos melhores vestibulares do pa\u00eds. <\/p><\/blockquote>\n\n\n\n<p>Quando voc\u00ea terminar as quest\u00f5es sobre<strong> Hip\u00e9rbole<\/strong>, fa\u00e7a nosso <a href=\"https:\/\/beduka.com\/simulado-enem\/#\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Simulado Enem<\/a>. Coloque em pr\u00e1tica todo o seu estudo, escolhendo as mat\u00e9rias que deseja treinar.<\/p>\n\n\n\n<p>As Hip\u00e9rboles s\u00e3o estudadas por um campo da matem\u00e1tica denominado de <a href=\"https:\/\/es.wikipedia.org\/wiki\/Geometr%C3%ADa_anal%C3%ADtica\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Geometria Anal\u00edtica<\/a><\/p>\n\n\n\n<p>A Geometria Anal\u00edtica estuda o objeto matem\u00e1tico, figura geom\u00e9trica, forma, entre outros, que possam ser representados no espa\u00e7o geometricamente por um desenho ou algebricamente por uma f\u00f3rmula. <strong>A geometria anal\u00edtica une a geometria com a \u00e1lgebra, permitindo a cria\u00e7\u00e3o de f\u00f3rmulas matem\u00e1ticas.<\/strong><\/p>\n\n\n\n<p><strong>Neste post aprenderemos a resolver quest\u00f5es envolvendo Hip\u00e9rbole<\/strong>, que s\u00e3o figuras geom\u00e9tricas espec\u00edficas, e ao final voc\u00ea colocar\u00e1 seu conhecimento em pr\u00e1tica com exerc\u00edcios sobre Hip\u00e9rbole.<\/p>\n\n\n\n<ul><li>Aproveita tamb\u00e9m a <a href=\"https:\/\/beduka.com\/blog\/exercicios\/matematica-exercicios\/exercicios-de-geometria-analitica\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">lista de exerc\u00edcios sobre Geometria Anal\u00edtica<\/a> que fizemos para voc\u00ea!<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">O que \u00e9 Hip\u00e9rbole?<\/h2>\n\n\n\n<p>Considere F\u00b9 e F2\u00b2 como sendo dois pontos diferentes do plano e 2c a dist\u00e2ncia existente entre eles.<strong> A Hip\u00e9rbole \u00e9 o conjunto dos pontos do plano, onde a diferen\u00e7a, em valor absoluto, das dist\u00e2ncias \u00e0 F\u00b9 e F\u00b2 \u00e9 a constante 2a (0 &lt;2a &lt;2c).<\/strong><\/p>\n\n\n\n<p>Para conseguir entender o par\u00e1grafo anterior e os pr\u00f3ximos t\u00f3picos, veja a figura abaixo:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/hip\u00e9rbole-imagem-1.jpg\" alt=\"Exemplo de Hip\u00e9rbole\" class=\"wp-image-17652\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/hip\u00e9rbole-imagem-1.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/hip\u00e9rbole-imagem-1-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/hip\u00e9rbole-imagem-1-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Quais s\u00e3o os elementos da Hip\u00e9rbole?<\/h3>\n\n\n\n<ul><li><strong>Focos: <\/strong>s\u00e3o os pontos F\u00b9 e F\u00b2;<\/li><li><strong>Dist\u00e2ncia focal: <\/strong>dist\u00e2ncia entre os dois focos denominada de 2c;<\/li><li><strong>Centro: <\/strong>ponto m\u00e9dio do segmento criado pela dist\u00e2ncia entre os dois focos;<\/li><li><strong>Eixo real: <\/strong>A hip\u00e9rbole intercepta o segmento F\u00b9F\u00b2 nos pontos A\u00b9 e A\u00b2. o segmento gerado pela dist\u00e2ncia entre A\u00b9 e A\u00b2 \u00e9 denominado de eixo real, que possui comprimento igual a 2a<\/li><li><strong>Eixo imagin\u00e1rio: <\/strong>\u00e9 o segmento de reta B\u00b9B\u00b2 perpendicular ao eixo real, com ponto m\u00e9dio no centro da hip\u00e9rbole. O comprimento do eixo imagin\u00e1rio \u00e9 igual a 2b;<\/li><li><strong>Excentricidade:<\/strong> \u00e9 a raz\u00e3o a seguir.<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Quais s\u00e3o as equa\u00e7\u00f5es reduzidas da hip\u00e9rbole<\/h3>\n\n\n\n<p><strong>1\u00b0 caso &#8211; <\/strong>Onde a hip\u00e9rbole possui os focos sobre o eixo x e centro sobre a origem de um plano cartesiano:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_1.jpg\" alt=\"Qual a equa\u00e7\u00e3o da hip\u00e9rbole?\" class=\"wp-image-17653\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_1.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_1-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_1-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<p><strong>2\u00b0 caso &#8211; <\/strong>Onde a hip\u00e9rbole tamb\u00e9m possui centro na origem, mas seus focos est\u00e3o sobre o eixo y do plano cartesiano:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" loading=\"lazy\" width=\"600\" height=\"300\" src=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_2.jpg\" alt=\"Qual a equa\u00e7\u00e3o da hip\u00e9rbole?\" class=\"wp-image-17654\" srcset=\"https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_2.jpg 600w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_2-300x150.jpg 300w, https:\/\/beduka.com\/blog\/wp-content\/uploads\/2020\/08\/Hip\u00e9rbole_f\u00f3rmula_2-320x160.jpg 320w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Exerc\u00edcios sobre Hip\u00e9rbole com Gabarito<\/h2>\n\n\n\n<p>Que tal testar seu conhecimento com <strong>exerc\u00edcios sobre Hip\u00e9rbole? <\/strong>Assim voc\u00ea pode se certificar que entendeu esse conte\u00fado t\u00e3o valioso para o ENEM.&nbsp;<\/p>\n\n\n\n<p>Baixe <strong>gratuitamente<\/strong> o <a href=\"https:\/\/beduka.com\/blog\/dicas\/enem-dicas\/plano-de-estudos-gratuito\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Plano de Estudos<\/a> do <strong>Beduka<\/strong> e tenha uma prepara\u00e7\u00e3o<strong> perfeita <\/strong>para o <strong>ENEM<\/strong>.<\/p>\n\n\n\n<p>Ali\u00e1s, <strong>Parab\u00e9ns<\/strong> por ter lido at\u00e9 aqui. Voc\u00ea teve uma s\u00e1bia decis\u00e3o e <strong>est\u00e1 pronto para fazer as quest\u00f5es dos melhores vestibulares do pa\u00eds que separamos para voc\u00ea!<\/strong><\/p>\n\n\n\n<p><strong>Quest\u00e3o 1 &#8211; <a href=\"https:\/\/ufam.edu.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(UFAM)  (abre numa nova aba)\">(UFAM)<\/a><\/strong><a href=\"https:\/\/ufam.edu.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(UFAM)  (abre numa nova aba)\"> <\/a>Os pontos A(4, 0) e B(0, 6) s\u00e3o extremos de um di\u00e2metro da circunfer\u00eancia. Ent\u00e3o, a equa\u00e7\u00e3o reduzida da circunfer\u00eancia \u00e9:<\/p>\n\n\n\n<p><strong>a)<\/strong> x<sup>2<\/sup> + y<sup>2<\/sup> \u2013 6x \u2013 4y = 0<\/p>\n\n\n\n<p><strong>b)<\/strong> x<sup>2<\/sup> + y<sup>2<\/sup> \u2013 4x \u2013 6y = 0<\/p>\n\n\n\n<p><strong>c)<\/strong> x<sup>2<\/sup> + y<sup>2<\/sup> + 4x \u2013 6y = 0<\/p>\n\n\n\n<p><strong>d)<\/strong> x<sup>2<\/sup> + y<sup>2<\/sup> + 4x + 6y = 0<\/p>\n\n\n\n<p><strong>e)<\/strong> x<sup>2<\/sup> + y<sup>2<\/sup> \u2013 6x + 4y = 0<\/p>\n\n\n\n<p><strong>Quest\u00e3o 2 &#8211; <a href=\"https:\/\/www.puc-rio.br\/index.html\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(PUC-Rio) (abre numa nova aba)\">(PUC-Rio)<\/a><\/strong> O n\u00famero de pontos de intersec\u00e7\u00e3o das duas par\u00e1bolas y = x<sup>2<\/sup> e y = 2x<sup>2<\/sup> \u2013 1 \u00e9:<\/p>\n\n\n\n<p><strong>a)<\/strong> 0<\/p>\n\n\n\n<p><strong>b)<\/strong> 1<\/p>\n\n\n\n<p><strong>c)<\/strong> 2<\/p>\n\n\n\n<p><strong>d)<\/strong> 3<\/p>\n\n\n\n<p><strong>e)<\/strong> 4<\/p>\n\n\n\n<p><strong>Quest\u00e3o 3 &#8211; <a href=\"https:\/\/www.ifpb.edu.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(IFPB) (abre numa nova aba)\">(IFPB)<\/a> <\/strong>Dados dois pares ordenados (2,-4) e (2,0) que representam os v\u00e9rtices de um hip\u00e9rbole de foco (2, -2 + \u221a13), calcule a equa\u00e7\u00e3o da hip\u00e9rbole que satisfa\u00e7a as condi\u00e7\u00f5es dadas.<\/p>\n\n\n\n<p><strong>a) <\/strong>4x\u00b2 &#8211; 9y\u00b2 &#8211; 16x &#8211; 36y +16 = 0<\/p>\n\n\n\n<p><strong>b) <\/strong>2x\u00b2 &#8211; 3y\u00b2 &#8211; 16x &#8211; 36y + 16 = 0<\/p>\n\n\n\n<p><strong>c) <\/strong>4x\u00b2 &#8211; 9y\u00b2 + 16x &#8211; 36y &#8211; 16 = 0<\/p>\n\n\n\n<p><strong>d) <\/strong>4x\u00b2 \u2212 5y\u00b2 \u2212 16x &#8211; 36y + 16 = 0<\/p>\n\n\n\n<p><strong>e) <\/strong>4x\u00b2 &#8211; 16x &#8211; 36y + 16 = 0<\/p>\n\n\n\n<p><strong>Quest\u00e3o 4 &#8211; <a href=\"http:\/\/www.fumarc.com.br\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(FUMARC 2013) (abre numa nova aba)\">(FUMARC 2013)<\/a> <\/strong>Um artista recebeu uma encomenda para fazer um painel, esculpindo em uma chapa de a\u00e7o folhas e flores. Para determinar o formato do painel, o artista considerou a chapa de a\u00e7o como um plano cartesiano cujos eixos a dividiram em quatro quadrantes. Utilizou um segmento de reta e o deslocou nesse plano cartesiano, de tal forma que uma das extremidades permanecia sempre no eixo y e o seu ponto m\u00e9dio permanecia sempre no eixo x. Dessa maneira, o formato da figura desenhada pela outra extremidade \u00e9 uma<\/p>\n\n\n\n<p><strong>a)<\/strong> elipse.<\/p>\n\n\n\n<p><strong>b)<\/strong> par\u00e1bola.<\/p>\n\n\n\n<p><strong>c) <\/strong>hip\u00e9rbole.<\/p>\n\n\n\n<p><strong>d) <\/strong>circunfer\u00eancia.<\/p>\n\n\n\n<p><strong>Quest\u00e3o 5 &#8211; <a href=\"https:\/\/petrobras.com.br\/pt\/\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(Petrobr\u00e1s \u2013 Cesgranrio 2010)  (abre numa nova aba)\">(Petrobr\u00e1s \u2013 Cesgranrio 2010) <\/a><\/strong>Os v\u00e9rtices imagin\u00e1rios da hip\u00e9rbole de equa\u00e7\u00e3o abaixo s\u00e3o:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/EKBN1XiwiKYmYv_w8SiNTWk2pvvZTtsMCauaEdjO_Oc_jHimIiP_x2A052O38bohKlQgUlCihJh7Qjd6EGqjBBSVfIWYsCtIa-HwoCH7y2HFnxc_X8dA5QgWIkQt5rqYpt4IUXZf\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>a)<\/strong> (2,1) e (2,3)<\/p>\n\n\n\n<p><strong>b)<\/strong> (2,0) e (2,2)<\/p>\n\n\n\n<p><strong>c)<\/strong> (2,0) e (1,2)<\/p>\n\n\n\n<p><strong>d)<\/strong> (1,1) e (1,2)<\/p>\n\n\n\n<p><strong>e)<\/strong> (1,0) e (1,2)<\/p>\n\n\n\n<p><strong>Quest\u00e3o 6 &#8211; <a href=\"http:\/\/www.idecan.org.br\/Index.aspx\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(IDECAN)  (abre numa nova aba)\">(IDECAN)<\/a><\/strong><a href=\"http:\/\/www.idecan.org.br\/Index.aspx\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"(IDECAN)  (abre numa nova aba)\"> <\/a>Dados dois pares ordenados (2,-4) e (2,0) que representam os v\u00e9rtices de um hip\u00e9rbole de foco (2, -2 + \u221a13), calcule a equa\u00e7\u00e3o da hip\u00e9rbole que satisfa\u00e7a as condi\u00e7\u00f5es dadas.<\/p>\n\n\n\n<p><strong>a)<\/strong> 4<em>x<\/em><sup>2<\/sup> \u2013 9<em>y<\/em><sup>2<\/sup> \u2013 16<em>x<\/em> \u2013 36<em>y<\/em> +16 = 0<\/p>\n\n\n\n<p><strong>b)<\/strong> 2<em>x<\/em><sup>2<\/sup> \u2013 3<em>y<\/em><sup>2<\/sup> \u2013 16<em>x<\/em> \u2013 36<em>y<\/em> + 16 = 0<\/p>\n\n\n\n<p><strong>c)<\/strong> 4<em>x<\/em><sup>2<\/sup> \u2013 9<em>y<\/em><sup>2<\/sup> + 16<em>x<\/em> \u2013 36<em>y<\/em> \u2013 16 = 0<\/p>\n\n\n\n<p><strong>d)<\/strong> 4<em>x<\/em><sup>2<\/sup> \u2212 5<em>y<\/em><sup>2<\/sup> \u2212 16<em>x<\/em> \u2013 36<em>y<\/em> + 16 = 0<\/p>\n\n\n\n<p><strong>e)<\/strong> 4<em>x<\/em><sup>2 <\/sup>\u2013 16<em>x<\/em> \u2013 36<em>y<\/em> + 16 = 0<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Gabarito dos Exerc\u00edcios sobre Hip\u00e9rbole<\/h2>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 1 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>b)<\/strong> x<sup>2<\/sup> + y<sup>2<\/sup> \u2013 4x \u2013 6y = 0<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 2 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>c)<\/strong> 2<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 3 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> 4x\u00b2 &#8211; 9y\u00b2 &#8211; 16x &#8211; 36y +16 = 0<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 4 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> elipse.<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 5 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>e)<\/strong> (1,0) e (1,2)<\/p>\n\n\n\n<p><strong>Exerc\u00edcio resolvido da quest\u00e3o 6 &#8211;<\/strong><\/p>\n\n\n\n<p>Alternativa correta: <strong>a)<\/strong> 4<em>x<\/em><sup>2<\/sup> \u2013 9<em>y<\/em><sup>2<\/sup> \u2013 16<em>x<\/em> \u2013 36<em>y<\/em> +16 = 0<\/p>\n\n\n\n<p>Estude para o Enem com o <a href=\"https:\/\/beduka.com\/simulado\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\" (abre numa nova aba)\">Simulado Beduka.<\/a> <strong>\u00c9 gratuito!<\/strong><\/p>\n\n\n\n<p>Gostou dos nossos Exerc\u00edcios sobre <strong>Hip\u00e9rbole<\/strong>? Compartilhe com os seus amigos e comente abaixo sobre as \u00e1reas que voc\u00ea deseja mais explica\u00e7\u00f5es.<\/p>\n\n\n\n<p>Queremos te ajudar a encontrar a <strong>FACULDADE IDEAL! <\/strong>Logo abaixo, fa\u00e7a uma pesquisa por curso e cidade que te mostraremos todas as faculdades que podem te atender. <strong>Informamos a nota de corte, valor de mensalidade, nota do MEC, avalia\u00e7\u00e3o dos alunos, modalidades de ensino e muito mais.<\/strong><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hip\u00e9rboles adv\u00e9m de um corte que \u00e9 feito no cone (se\u00e7\u00e3o c\u00f4nica) onde formam-se duas \u201cmeia luas\u201d opostas. Hip\u00e9rbole \u00e9 conjunto de todos os pontos coplanares para os quais a diferen\u00e7a das dist\u00e2ncias a dois pontos fixos (focos) \u00e9 constante. Leia nosso resumo e fa\u00e7a nossa lista de exerc\u00edcios sobre Hip\u00e9rbole dos melhores vestibulares do [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":17656,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0},"categories":[610],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.10 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Exerc\u00edcios sobre Hip\u00e9rbole com Gabarito<\/title>\n<meta name=\"description\" content=\"Lista de exerc\u00edcios sobre Hip\u00e9rbole selecionados dos melhores vestibulares do pa\u00eds. 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