{"id":6805,"date":"2021-10-21T17:44:25","date_gmt":"2021-10-21T12:14:25","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6805"},"modified":"2021-10-25T00:04:39","modified_gmt":"2021-10-24T18:34:39","slug":"find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/","title":{"rendered":"Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Let \\(2x^2 + 5xy + 2y^2 + 4x + 5y + k\\) = 0 be asymptotes. This will represent two straight line<\/p>\n<p>so \\(abc + 2fgh &#8211; af^2 &#8211; bg^2 &#8211; ch^2\\) = 0 \\(\\implies\\) 4k + 25 &#8211; \\(25\\over 2\\) &#8211; 8 &#8211; \\(25\\over 4\\)k = 0<\/p>\n<p>\\(\\implies\\) k = 2<\/p>\n<p>\\(\\implies\\) \\(2x^2 + 5xy + 2y^2 + 4x + 5y + 2\\) = 0 are asymptotes<\/p>\n<p>\\(\\implies\\) (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes<\/p>\n<p>and\u00a0 \\(2x^2 + 5xy + 2y^2 + 4x + 5y + c\\) = 0 is general equation of hyperbola.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\">Similar Questions<\/h3>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/angle-between-asymptotes-of-hyperbola-xy8-is\/\" aria-label=\"\u201cAngle between asymptotes of hyperbola xy=8 is\u201d (Edit)\">Angle between asymptotes of hyperbola xy=8 is<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/find-the-normal-to-the-hyperbola-x2over-16-y2over-9-1-whose-slope-is-1\/\" aria-label=\"\u201cFind the normal to the hyperbola \\(x^2\\over 16\\) \u2013 \\(y^2\\over 9\\) = 1 whose slope is 1.\u201d (Edit)\">Find the normal to the hyperbola \\(x^2\\over 16\\) \u2013 \\(y^2\\over 9\\) = 1 whose slope is 1.<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/find-the-equation-of-the-tangent-to-the-hyperbola-x2-4y2-36-which-is-perpendicular-to-the-line-x-y-4-0\/\" aria-label=\"\u201cFind the equation of the tangent to the hyperbola \\(x^2 \u2013 4y^2\\) = 36 which is perpendicular to the line x \u2013 y + 4 = 0\u201d (Edit)\">Find the equation of the tangent to the hyperbola \\(x^2 \u2013 4y^2\\) = 36 which is perpendicular to the line x \u2013 y + 4 = 0<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/the-eccentricity-of-the-conjugate-hyperbola-to-the-hyperbola-x2-3y2-1-is\/\" aria-label=\"\u201cThe eccentricity of the conjugate hyperbola to the hyperbola \\(x^2-3y^2\\) = 1 is\u201d (Edit)\">The eccentricity of the conjugate hyperbola to the hyperbola \\(x^2-3y^2\\) = 1 is<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/if-the-foci-of-a-hyperbola-are-foci-of-the-ellipse-x2over-25-y2over-9-1-if-the-eccentricity-of-the-hyperbola-be-2-then-its-equation-is\/\" aria-label=\"\u201cIf the foci of a hyperbola are foci of the ellipse \\(x^2\\over 25\\) + \\(y^2\\over 9\\) = 1. If the eccentricity of the hyperbola be 2, then its equation is :\u201d (Edit)\">If the foci of a hyperbola are foci of the ellipse \\(x^2\\over 25\\) + \\(y^2\\over 9\\) = 1. If the eccentricity of the hyperbola be 2, then its equation is :<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Let \\(2x^2 + 5xy + 2y^2 + 4x + 5y + k\\) = 0 be asymptotes. This will represent two straight line so \\(abc + 2fgh &#8211; af^2 &#8211; bg^2 &#8211; ch^2\\) = 0 \\(\\implies\\) 4k + 25 &#8211; \\(25\\over 2\\) &#8211; 8 &#8211; \\(25\\over 4\\)k = 0 \\(\\implies\\) k = 2 \\(\\implies\\) &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/\"> <span class=\"screen-reader-text\">Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[51,43],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.\" \/>\n<meta property=\"og:description\" content=\"Solution : Let (2x^2 + 5xy + 2y^2 + 4x + 5y + k) = 0 be asymptotes. This will represent two straight line so (abc + 2fgh &#8211; af^2 &#8211; bg^2 &#8211; ch^2) = 0 (implies) 4k + 25 &#8211; (25over 2) &#8211; 8 &#8211; (25over 4)k = 0 (implies) k = 2 (implies) &hellip; Find the asymptotes of the hyperbola (2x^2 + 5xy + 2y^2 + 4x + 5y) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes. Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-21T12:14:25+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-24T18:34:39+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Find the asymptotes of the hyperbola \\\\(2x^2 + 5xy + 2y^2 + 4x + 5y\\\\) = 0. 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Find also the general equation of all the hyperbolas having the same set of asymptotes.\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/","og_locale":"en_US","og_type":"article","og_title":"Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.","og_description":"Solution : Let (2x^2 + 5xy + 2y^2 + 4x + 5y + k) = 0 be asymptotes. This will represent two straight line so (abc + 2fgh &#8211; af^2 &#8211; bg^2 &#8211; ch^2) = 0 (implies) 4k + 25 &#8211; (25over 2) &#8211; 8 &#8211; (25over 4)k = 0 (implies) k = 2 (implies) &hellip; Find the asymptotes of the hyperbola (2x^2 + 5xy + 2y^2 + 4x + 5y) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes. Read More &raquo;","og_url":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/","og_site_name":"Mathemerize","article_published_time":"2021-10-21T12:14:25+00:00","article_modified_time":"2021-10-24T18:34:39+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.","datePublished":"2021-10-21T12:14:25+00:00","dateModified":"2021-10-24T18:34:39+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/"},"wordCount":169,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"articleSection":["Hyperbola Questions","Maths Questions"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/","url":"https:\/\/mathemerize.com\/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes\/","name":"Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0. 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