{"id":3269,"date":"2021-07-21T17:01:53","date_gmt":"2021-07-21T17:01:53","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3269"},"modified":"2021-11-23T21:41:55","modified_gmt":"2021-11-23T16:11:55","slug":"equation-of-normal-to-ellipse-in-all-forms","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-normal-to-ellipse-in-all-forms\/","title":{"rendered":"Equation of Normal to Ellipse in all Forms"},"content":{"rendered":"

Equation of Normal to ellipse : \\(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 ellipse at (\\(x_1, y_1\\)) 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 normal to the ellipse \\(9x^2+16y^2\\) = 288 at the point (4,3).<\/p>\n

Solution : <\/span>We have, \\(9x^2+16y^2\\) = 288

\nComparing with general equation of ellipse,

\n\\(a^2\\) = 32 and \\(b^2\\) = 18

\nThe normal to given ellipse in point form is \\(a^2x\\over x_1\\) + \\(b^2y\\over y_1\\) = \\(a^2-b^2\\)

\nSo, \\(32x\\over 4\\) + \\(18y\\over 3\\) = 14

\n 8x + 6y = 14

\nHence, normal to given ellipse is 8x + 6y = 14.<\/p>\n\n\n

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

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

\n

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

Example : <\/span> Find the normal to the ellipse \\(x^2 + 2y^2\\) = 6 whose slope is 2.<\/p>\n

Solution : <\/span>We have, \\(x^2 + 2y^2\\) = 6

\nComparing with \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1

\n\\(a^2\\) = 6 and \\(b^2\\) = 3

\nThe normal to given ellipse in slope form is y = mx \\(\\mp\\) \\({(a^2-b^2)m}\\over \\sqrt{a^2 + b^2m^2}\\)

\nSo, y = 2x \\(\\mp\\) \\({3m}\\over \\sqrt{18}\\)

\nHence, normal to given ellipse isy = 2x \\(\\mp\\) \\({3m}\\over \\sqrt{18}\\).<\/p>\n\n\n

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

The equation of normal to the given ellipse at its point (acos\\(\\theta\\), bsin\\(\\theta\\)), is<\/p>\n

\n

\\(axsec\\theta \\) – \\(bycosec\\theta \\) = \\(a^2-b^2\\)<\/p>\n

or \\(ax\\over cos\\theta\\) – \\(by\\over sin\\theta\\) = \\(a^2-b^2\\)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the condition that the line lx + my = n may be a normal to the ellipse \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1<\/p>\n

Solution : <\/span>Equation of normal to the given ellipse at its point (acos\\(\\theta\\), bsin\\(\\theta\\)), is

\n\\(ax\\over cos\\theta\\) – \\(by\\over sin\\theta\\) = \\(a^2-b^2\\) ….(i)

\nIf line lx + my = n is also normal to the ellipse then there must be a value of \\(\\theta\\) for which line (i) and line lx + my = n are identical. for that value of \\(\\theta\\) we have

\n\\(l\\over ({a\\over cos\\theta})\\) = \\(m\\over -({b\\over sin\\theta})\\) = \\(n\\over {l(a^2 – b^2)}\\) or

\n\\(\\cos\\theta\\) = \\(an\\over {l(a^2 – b^2)}\\) ……(iii)

\n\\(\\sin\\theta\\) = \\(-bn\\over {m(a^2 – b^2)}\\) ……(iv)

\nSquaring and adding (iii) and (iv), we get\n\\(n^2\\over {(a^2 – b^2)^2}\\) (\\({a^2\\over l^2} + {b^2\\over m^2}\\)) which is the required condition.<\/p>\n\n\n

Hope you learnt equation of normal to ellipse in all forms, learn more concepts of ellipse and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

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

Equation of Normal to ellipse : \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1 (a) Point form : The Equation of normal to the given ellipse at (\\(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 normal to the ellipse \\(9x^2+16y^2\\) = 288 at the point (4,3). Solution : …<\/p>\n

Equation of Normal to Ellipse 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":[24],"tags":[349,351,352,350],"yoast_head":"\nEquation of Normal to Ellipse in all Forms - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn what is the equation of normal to ellipse in all forms i.e, point form, slope form and parametric form.\" \/>\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-ellipse-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 Ellipse in all Forms - 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