{"id":3260,"date":"2021-07-21T17:01:58","date_gmt":"2021-07-21T17:01:58","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3260"},"modified":"2021-11-23T21:40:24","modified_gmt":"2021-11-23T16:10:24","slug":"equation-of-tangent-to-ellipse-in-all-forms","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-tangent-to-ellipse-in-all-forms\/","title":{"rendered":"Equation of Tangent to Ellipse in all Forms"},"content":{"rendered":"

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

(a) Point form :<\/h2>\n

The equation of tangent to the given ellipse at its point (\\(x_1, y_1\\)) is\u00a0<\/p>\n

\n

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

Note – <\/strong>For general ellipse replace \\(x^2\\) by \\(xx_1\\), \\(y^2\\) by \\(yy_1\\), 2x by \\(x + x_1\\), 2y by \\(y + y_1\\), 2xy by \\(xy_1 + yx_1\\) and c by c.<\/p>\n

(b) Slope form :<\/h2>\n

The Equation of tangent to the given ellipse whose slope is ‘m<\/strong>‘, is<\/p>\n

\n

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

Point of contact are (\\({\\pm} a^2m\\over \\sqrt{a^2m^2 + b^2}\\), \\({\\pm} b^2\\over \\sqrt{a^2m^2 + b^2}\\)).<\/p>\n<\/blockquote>\n

Note that there are two tangents to the ellipse having the same m, i.e. there are two tangents parallel to any given direction.<\/p>\n

(c) Parametric form :<\/h2>\n

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

\n

\\(xcos\\theta\\over a\\) + \\(ysin\\theta\\over b\\) = 1<\/p>\n<\/blockquote>\n

Note<\/strong> – The point of the intersection of the tangents at the point \\(\\alpha\\) & \\(\\beta\\) is (a\\(cos{\\alpha+\\beta\\over 2}\\over cos{\\alpha -\\beta\\over 2}\\), b\\(sin{\\alpha+\\beta\\over 2}\\over cos{\\alpha-\\beta\\over 2}\\))<\/p>\n\n\n

Example : <\/span> Find the equation of the tangents to the ellipse \\(3x^2+4y^2\\) = 12 which are perpendicular to the line y + 2x = 4<\/p>\n

Solution : <\/span>Let m be the slope of the tangent, since the tangent is perpendicular to the line y + 2x = 4

\\(\\therefore\\)   mx – 2 = -1 \\(\\implies\\) m = \\(1\\over 2\\)

Since \\(3x^2+4y^2\\) = 12 or \\(x^2\\over 4\\) + \\(y^2\\over 3\\) = 1

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

\\(\\therefore\\)   \\(a^2\\) = 4 and \\(b^2\\) = 3

So the equation of the tangent are y = \\(1\\over 2\\)x \\(\\pm\\) \\(\\sqrt{4\\times {1\\over 4} + 3}\\)

\\(\\implies\\) y = \\(1\\over 2\\)x \\(\\pm\\) 2 or x – 2y \\(\\pm\\) 4 = 0

<\/p>\n\n\n

Hope you learnt equation of tangent 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
Next – Equation of Normal to Ellipse in all Forms<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Different Types of Ellipse \u2013 Parametric Equation of Ellipse<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Equation of Tangent to Ellipse \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1 : (a) Point form : The equation of tangent to the given ellipse at its point (\\(x_1, y_1\\)) is\u00a0 \\(x{x_1}\\over a^2\\) + \\(y{y_1}\\over b^2\\) = 1. Note – For general ellipse replace \\(x^2\\) by \\(xx_1\\), \\(y^2\\) by \\(yy_1\\), 2x by \\(x + x_1\\), …<\/p>\n

Equation of Tangent 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":[345,346,348,347],"yoast_head":"\nEquation of Tangent to Ellipse in all Forms - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn what is the equation of tangent 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-tangent-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 Tangent to Ellipse in all Forms - 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