{"id":3280,"date":"2021-07-21T17:02:12","date_gmt":"2021-07-21T17:02:12","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3280"},"modified":"2021-11-24T23:25:54","modified_gmt":"2021-11-24T17:55:54","slug":"equation-of-tangent-to-hyperbola-in-all-forms","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/","title":{"rendered":"Equation of Tangent to Hyperbola in all Forms"},"content":{"rendered":"

Equation of Tangent to Hyperbola \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1<\/h2>\n

(a) Point form :<\/h3>\n

The equation of tangent to the given hyperbola at its point (\\(x_1, y_1\\))<\/strong> is<\/p>\n

\n

\\(x{x_1}\\over a^2\\) – \\(y{y_1}\\over b^2\\) = 1<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the equation of tangent to the hyperbola \\(16x^2\\) – \\(9y^2\\) = 144 at (5, 16\/3).<\/p>\n

Solution : <\/span>We have, \\(16x^2\\) – \\(9y^2\\) = 144

\n\\(\\implies\\) \\(x^2\\over 9\\) – \\(y^2\\over 16\\) = 1

\nCompare given equation with \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1

\na = 3 and b = 16

\nHence, required equation of tangent is \\(5x\\over 9\\) – \\(16\/3\\over 16\\) = 1

\n= \\(5x\\over 9\\) – \\(y\\over 3\\) = 1

<\/p>\n\n\n

(b) Slope form :<\/h3>\n

The equation of tangent to the given hyperbola whose slope is ‘m’, is<\/p>\n

\n

y = mx \\(\\pm\\) \\(\\sqrt{a^2m^2 – b^2}\\)<\/p>\n

The Point of contact are (\\({\\mp} a^2m\\over \\sqrt{a^2m^2 – b^2}\\), \\({\\mp} b^2\\over \\sqrt{a^2m^2 – b^2}\\))<\/p>\n

Note that there are two parallel tangents having the same slope m.<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the tangent to the hyperbola \\(x^2\\over 25\\) – \\(y^2\\over 16\\) = 1 whose slope is 1.<\/p>\n

Solution : <\/span>We have, \\(x^2\\over 25\\) – \\(y^2\\over 16\\) = 1

\nCompare given equation with \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1

\na = 5 and b = 4

\nHence, required equation of normal is y = x \\(\\pm\\) \\(\\sqrt{9}\\)

\n\\(\\implies\\) y = x \\(\\pm\\) 3

<\/p>\n\n\n

(c) Parametric form :<\/h3>\n

The equation of tangent to the given hyperbola at the point (asec\\(\\theta\\), btan\\(\\theta\\))<\/strong>, is<\/p>\n

\n

\\(xsec\\theta\\over a\\) – \\(ytan\\theta\\over b\\) = 1<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find 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\n

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

Equation of Tangent to Hyperbola \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1 (a) Point form : The equation of tangent to the given hyperbola at its point (\\(x_1, y_1\\)) is \\(x{x_1}\\over a^2\\) – \\(y{y_1}\\over b^2\\) = 1 Example : Find the equation of tangent to the hyperbola \\(16x^2\\) – \\(9y^2\\) = 144 at (5, 16\/3). …<\/p>\n

Equation of Tangent to Hyperbola in all Forms<\/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":[389,390,393,394,385],"yoast_head":"\nEquation of Tangent to Hyperbola in all Forms - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn equation of tangent to hyperbola in all forms i.e. point form, slope form and parametric form with example,\" \/>\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\/equation-of-tangent-to-hyperbola-in-all-forms\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Tangent to Hyperbola in all Forms - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn equation of tangent to hyperbola in all forms i.e. point form, slope form and parametric form with example,\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-07-21T17:02:12+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-24T17:55:54+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=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Equation of Tangent to Hyperbola in all Forms\",\"datePublished\":\"2021-07-21T17:02:12+00:00\",\"dateModified\":\"2021-11-24T17:55:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/\"},\"wordCount\":351,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"equation of tangent to hyperbola\",\"equation of tangent to hyperbola in parametric form\",\"equation of tangent to hyperbola in point form\",\"equation of tangent to hyperbola in slope form\",\"hyperbola\"],\"articleSection\":[\"Hyperbola\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/\",\"url\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/\",\"name\":\"Equation of Tangent to Hyperbola in all Forms - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-07-21T17:02:12+00:00\",\"dateModified\":\"2021-11-24T17:55:54+00:00\",\"description\":\"In this post you will learn equation of tangent to hyperbola in all forms i.e. point form, slope form and parametric form with example,\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Equation of Tangent to Hyperbola in all Forms\"}]},{\"@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":"Equation of Tangent to Hyperbola in all Forms - Mathemerize","description":"In this post you will learn equation of tangent to hyperbola in all forms i.e. point form, slope form and parametric form with example,","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\/equation-of-tangent-to-hyperbola-in-all-forms\/","og_locale":"en_US","og_type":"article","og_title":"Equation of Tangent to Hyperbola in all Forms - Mathemerize","og_description":"In this post you will learn equation of tangent to hyperbola in all forms i.e. point form, slope form and parametric form with example,","og_url":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/","og_site_name":"Mathemerize","article_published_time":"2021-07-21T17:02:12+00:00","article_modified_time":"2021-11-24T17:55:54+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Equation of Tangent to Hyperbola in all Forms","datePublished":"2021-07-21T17:02:12+00:00","dateModified":"2021-11-24T17:55:54+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/"},"wordCount":351,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["equation of tangent to hyperbola","equation of tangent to hyperbola in parametric form","equation of tangent to hyperbola in point form","equation of tangent to hyperbola in slope form","hyperbola"],"articleSection":["Hyperbola"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/","url":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/","name":"Equation of Tangent to Hyperbola in all Forms - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-07-21T17:02:12+00:00","dateModified":"2021-11-24T17:55:54+00:00","description":"In this post you will learn equation of tangent to hyperbola in all forms i.e. point form, slope form and parametric form with example,","breadcrumb":{"@id":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/equation-of-tangent-to-hyperbola-in-all-forms\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Equation of Tangent to Hyperbola in all Forms"}]},{"@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\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3280"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=3280"}],"version-history":[{"count":14,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3280\/revisions"}],"predecessor-version":[{"id":8575,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3280\/revisions\/8575"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=3280"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=3280"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=3280"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}