{"id":5997,"date":"2021-10-04T21:27:51","date_gmt":"2021-10-04T15:57:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5997"},"modified":"2021-10-04T21:28:40","modified_gmt":"2021-10-04T15:58:40","slug":"hyperbola-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/hyperbola-examples\/","title":{"rendered":"Hyperbola Examples"},"content":{"rendered":"

Here you will learn some hyperbola examples for better understanding of hyperbola concepts<\/a><\/span>.<\/p>\n\n\n

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\n\n \n

Example 1 : <\/span> 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 : <\/p>\n

Solution : <\/span>For ellipse e = \\(4\\over 5\\), so foci = (\\(\\pm\\)4, 0)

for hyperbola e = 2, so a = \\(ae\\over e\\) = \\(4\\over 2\\) = 2, b = \\(2\\sqrt{4-1}\\) = 2\\(\\sqrt{3}\\)

Hence the equation of the hyperbola is \\(x^2\\over 4\\) – \\(y^2\\over 12\\) = 1<\/p>\n


\n\n \n

Example 2 : <\/span> The eccentricity of the conjugate hyperbola to the hyperbola \\(x^2-3y^2\\) = 1 is-<\/p>\n

Solution : <\/span>Equation of the conjugate hyperbola to the hyperbola \\(x^2-3y^2\\) = 1 is

\\(-x^2-3y^2\\) = 1 \\(\\implies\\) \\(-x^2\\over 1\\) + \\(y^2\\over {1\/3}\\) = 1

Here \\(a^2\\) = 1, \\(b^2\\) = \\(1\\over 3\\)

\\(\\therefore\\)   eccentricity e = \\(\\sqrt{1 + a^2\/b^2}\\) = \\(\\sqrt{1+3}\\) = 2\n <\/p>


\n\n \n

Example 3 : <\/span> Find the equation of the tangent to the hyperbola \\(x^2 – 4y^2\\) = 36 which is perpendicular to the line x – y + 4 = 0<\/p>\n

Solution : <\/span>Let m be the slope of the tangent, since the tangent is perpendicular to the line x – y = 0

\\(\\therefore\\)   m\\(\\times\\)1 = -1 \\(\\implies\\) m = -1

Since \\(x^2-4y^2\\) = 36 or \\(x^2\\over 36\\) – \\(y^2\\over 9\\) = 1

Comparing this with \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1

\\(\\therefore\\)   \\(a^2\\) = 36 and \\(b^2\\) = 9

So the equation of the tangent are y = -1x \\(\\pm\\) \\(\\sqrt{36\\times {-1}^2 – 9}\\)

\\(\\implies\\) y = x \\(\\pm\\) \\(\\sqrt{27}\\) \\(\\implies\\) x + y \\(\\pm\\) 3\\(\\sqrt{3}\\) = 0<\/p>\n


\n\n

Example 4 : <\/span> 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.<\/p>\n

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

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

\\(\\implies\\) k = 2

\\(\\implies\\) \\(2x^2 + 5xy + 2y^2 + 4x + 5y + 2\\) = 0 are asymptotes

\\(\\implies\\) (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes

and   \\(2x^2 + 5xy + 2y^2 + 4x + 5y + c\\) = 0 is general equation of hyperbola.<\/p>\n
\n

Practice these given hyperbola examples to test your knowledge on concepts of hyperbola.<\/p>\n \n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some hyperbola examples for better understanding of hyperbola concepts. Example 1 : 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 : Solution : For ellipse e = \\(4\\over 5\\), …<\/p>\n

Hyperbola Examples<\/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":[],"yoast_head":"\nHyperbola Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these Hyperbola examples to practice and test your knowledge of Hyperbola concepts.\" \/>\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\/hyperbola-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hyperbola Examples - 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