{"id":3322,"date":"2021-07-22T11:33:46","date_gmt":"2021-07-22T11:33:46","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3322"},"modified":"2021-11-27T18:14:07","modified_gmt":"2021-11-27T12:44:07","slug":"equation-of-tangent-to-parabola","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-tangent-to-parabola\/","title":{"rendered":"Equation of Tangent to Parabola in all Forms"},"content":{"rendered":"

The equation of tangent to parabola in point form, slope form and parametric form are given below with examples.<\/p>\n

Condition of Tangency for Parabola :<\/h2>\n

(a)\u00a0 The line y = mx + c meets the parabola \\(y^2\\) = 4ax in two points real, coincident or imaginary according as a >=< cm \\(\\implies\\) condition of tangency<\/strong> is,<\/p>\n

\n

\u00a0Condition of tangency is \u00a0c = \\(a\\over m\\)<\/p>\n<\/blockquote>\n

Note :<\/strong> Line y = mx + c will be tangent to parabola \\(x^2\\) = 4ay if c = -a\\(m^2\\)<\/p>\n

(b) Length of the chord intercepted by the parabola \\(y^2\\) = 4ax on the line y = mx + c is \\(4\\over m^2\\)\\(\\sqrt{a(1+m^2)(a-mc)}\\)<\/p>\n

Note :<\/strong> Length of the focal chord making an angle \\(\\alpha\\) with the x-axis is 4a \\({cosec}^2\\alpha\\).<\/p>\n

Equation of Tangent to Parabola \\(y^2 = 4ax\\)<\/h2>\n

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

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

\n

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

Example : <\/span> Find the tangent to the parabola \\(y^2 = 16x\\) at (5, 2).<\/p>\n

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

\nCompare given equation with \\(y^2 = 4ax\\)

\n\\(\\implies\\) a = 4

\nHence, required equation of tangent is 2y = 8(x + 5)

\n= 2y = 8x + 40

<\/p>\n\n\n

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

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

\n

y = mx + \\(a\\over m\\), (m \\(\\ne\\) 0)<\/p>\n

The point of contact is (\\(a\\over m^2\\), \\(2a\\over m\\))<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the tangent to the parabola \\(y^2 = 8x\\) whose slope is 3.<\/p>\n

Solution : <\/span>We have, \\(y^2 = 8x\\)

\nCompare given equation with \\(y^2 = 4ax\\)

\na = 2

\nHence, required equation of tangent is y = 3x + \\(2\\over 3\\)

<\/p>\n\n\n

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

The tangent to the given parabola at its point P(t), is<\/p>\n

\n

ty = x + a\\(t^2\\)<\/p>\n

Note –<\/strong> Point of intersection of the tangents at the points \\(t_1\\) & \\(t_2\\) is [a\\(t_1t_2\\), a(\\(t_1 + t_2\\))].<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the equation of the tangents to the parabola \\(y^2\\) = 9x which go through the point (4,10).<\/p>\n

Solution : <\/span>tangent to the parabola \\(y^2\\) = 9x is

\ny = mx + \\(9\\over 4m\\)

\nSince it passes through (4,10)

\n\\(\\therefore\\) 10 = 4m + \\(9\\over 4m\\) \\(\\implies\\) 16\\(m^2\\) – 40m + 9 = 0

\nm = \\(1\\over 4\\), \\(9\\over 4\\)

\n\\(\\therefore\\) Equation of tangent’s are y = \\(x\\over 4\\) + 9 & y = \\(9x\\over 4\\) + 1

<\/p>\n\n\n

Hope you learnt equation of tangent to parabola in point form, slope form and parametric form, learn more concepts of parabola <\/a><\/span>and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – Equation of Normal to Parabola in all Forms<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Different Types of Parabola Equations<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

The equation of tangent to parabola in point form, slope form and parametric form are given below with examples. Condition of Tangency for Parabola : (a)\u00a0 The line y = mx + c meets the parabola \\(y^2\\) = 4ax in two points real, coincident or imaginary according as a >=< cm \\(\\implies\\) condition of tangency …<\/p>\n

Equation of Tangent to Parabola 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":[12],"tags":[526,522,525,523,524],"yoast_head":"\nEquation of Tangent to Parabola in all Forms - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn equation of tangent to parabola 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-parabola\/\" \/>\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 Parabola in all Forms - 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