{"id":3282,"date":"2021-07-21T17:02:02","date_gmt":"2021-07-21T17:02:02","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3282"},"modified":"2021-11-24T23:27:04","modified_gmt":"2021-11-24T17:57:04","slug":"equation-of-normal-to-hyperbola-in-all-forms","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-normal-to-hyperbola-in-all-forms\/","title":{"rendered":"Equation of Normal to Hyperbola in all Forms"},"content":{"rendered":"

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

(a) Point form :<\/h3>\n

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

\n

\\(a^2x\\over x_1\\) + \\(b^2y\\over y_1\\) = \\(a^2+b^2\\) = \\(a^2e^2\\)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the equation of normal to the hyperbola \\(x^2\\over 25\\) – \\(y^2\\over 16\\) = 1 at (5, 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 \\(25x\\over 5\\) + \\(16y\\over 1\\) = 41

\n= 5x + 16y = 41

<\/p>\n\n\n

(b) Slope form :<\/h3>\n

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

\n

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

Foot of normal are (\\(\\mp\\)\\({a^2}\\over \\sqrt{a^2 – m^2b^2}\\), \\(\\mp\\)\\({mb^2}\\over \\sqrt{a^2 – m^2b^2}\\)).<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the normal to the hyperbola \\(x^2\\over 16\\) – \\(y^2\\over 9\\) = 1 whose slope is 1.<\/p>\n

Solution : <\/span>We have, \\(x^2\\over 16\\) – \\(y^2\\over 9\\) = 1

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

\na = 4 and b = 3

\nHence, required equation of normal is y = x \\(\\mp\\) \\({25}\\over \\sqrt{7}\\)

\n<\/p>\n\n\n

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

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

\n

\\(ax\\over sec\\theta \\) + \\(by\\over tan\\theta \\) = \\(a^2+b^2\\) = \\(a^2e^2\\)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Line \\(xcos\\alpha\\) + \\(ysin\\alpha\\) = p is a normal to the hyperbola \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1, if<\/p>\n

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

\nThe normal to hyperbola is \\(ax\\over sec\\theta \\) + \\(by\\over tan\\theta \\) = \\(a^2+b^2\\)

\ncomparing it with the given line equation

\n\\(acos\\theta\\over cos\\alpha\\) = \\(bcot\\theta\\over sin\\alpha\\) = \\(a^2 + b^2\\over p\\)

\n\\(\\implies\\) \\(sec\\theta\\) = \\(ap\\over cos\\alpha(a^2+b^2)\\), \\(tan\\theta\\) = \\(bp\\over sin\\alpha(a^2+b^2)\\)

\nEliminating \\(\\theta\\), we get

\n\\(a^2sec^2\\alpha\\) – \\(b^2cosec^2\\alpha\\) = \\((a^2 + b^2)^2)\\over p^2\\)

<\/p>\n\n\n\n

\n
Next – Equation of Asymptotes of Hyperbola \u2013 Director Circle<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Equation of Tangent to Hyperbola in all Forms<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Equation of Normal to hyperbola : \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1 (a) Point form : The equation of normal to the given hyperbola at the point P(\\(x_1, y_1\\)) is \\(a^2x\\over x_1\\) + \\(b^2y\\over y_1\\) = \\(a^2+b^2\\) = \\(a^2e^2\\) Example : Find the equation of normal to the hyperbola \\(x^2\\over 25\\) – \\(y^2\\over 16\\) = …<\/p>\n

Equation of Normal 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":[395,396,392,391,385],"yoast_head":"\nEquation of Normal to Hyperbola in all Forms - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn equation of normal 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-normal-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 Normal to Hyperbola in all Forms - 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