{"id":3856,"date":"2021-08-09T14:20:58","date_gmt":"2021-08-09T14:20:58","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3856"},"modified":"2021-11-24T23:07:37","modified_gmt":"2021-11-24T17:37:37","slug":"asymptotes-of-hyperbola","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/asymptotes-of-hyperbola\/","title":{"rendered":"Equation of Asymptotes of Hyperbola – Director Circle"},"content":{"rendered":"

Here, you will learn equation of asymptotes of hyperbola and how to find the asymptotes of the hyperbola and the director circle of hyperbola.<\/p>\n

Let’s begin –<\/p>\n

Equation of Asymptotes of hyperbola<\/h2>\n

If the length of the perpendicular let fall from the point on the hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the\u00a0Asymptote of the hyperbola.<\/b><\/p>\n

How to find the asymptotes of the hyperbola :<\/strong><\/p>\n

Let y = mx + c is the asymptote of the hyperbola \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1.<\/p>\n

Solving these two we get the quadratic as \\((b^2 – a^2m^2)\\)\\(x^2\\) – 2\\(a^2\\)mcx – \\(a^2(b^2 + c^2)\\) = 0 …….(1)<\/p>\n

In order that y = mx + c be an asymptote, both roots of equation (1) must approach infinity, the condition for which are :<\/p>\n

coefficient of \\(x^2\\) = 0 & coefficient of x = 0.<\/p>\n

\\(\\implies\\) \\((b^2 – a^2m^2)\\) = 0 or m = \\(\\pm b\\over a\\) & \\(a^2\\)mc = 0 \\(\\implies\\) c = 0.<\/p>\n

\\(\\therefore\\)\u00a0 equations of asymptote are \\(x\\over a\\) + \\(y\\over b\\) = 0 and \\(x\\over a\\) – \\(y\\over b\\) = 0<\/p>\n

combined equation to the asymptotes \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 0<\/p>\n

When b = a, the asymptotes of the rectangular hyperbola.<\/p>\n

\\(x^2 – y^2\\) = \\(a^2\\) are y = \\(\\pm\\)x which are at right angles.<\/p>\n

Example<\/strong><\/span> : Find the asymptotes of the hyperbola \\(2x^2 + 5xy + 2y^2 + 4x + 5y\\) = 0<\/p>\n

Solution<\/strong><\/span> : Let \\(2x^2 + 5xy + 2y^2 + 4x + 5y + k\\) = 0 be asymptotes. This will represent two straight line<\/p>\n

so \\(abc + 2fgh – af^2 – bg^2 – ch^2\\) = 0 \\(\\implies\\) 4k + 25 – \\(25\\over 2\\) – 8 – \\(25\\over 4\\)k = 0<\/p>\n

\\(\\implies\\) k = 2<\/p>\n

\\(\\implies\\) \\(2x^2 + 5xy + 2y^2 + 4x + 5y + 2\\) = 0 are asymptotes<\/p>\n

\\(\\implies\\) (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes<\/p>\n\n\n

\n
Previous – Equation of Normal to Hyperbola in all Forms<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn equation of asymptotes of hyperbola and how to find the asymptotes of the hyperbola and the director circle of hyperbola. Let’s begin – Equation of Asymptotes of hyperbola If the length of the perpendicular let fall from the point on the hyperbola to a straight line tends to zero as the …<\/p>\n

Equation of Asymptotes of Hyperbola – Director Circle<\/span> Read More »<\/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":[25],"tags":[384,385],"yoast_head":"\nEquation of Asymptotes of Hyperbola - Director Circle<\/title>\n<meta name=\"description\" content=\"In this post, you will learn equation of asymptotes of hyperbola how to find the asymptotes of hyperbola and director circle of hyperbola.\" \/>\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\/asymptotes-of-hyperbola\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Asymptotes of Hyperbola - 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