{"id":3332,"date":"2021-07-22T11:34:12","date_gmt":"2021-07-22T11:34:12","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3332"},"modified":"2021-11-27T18:15:32","modified_gmt":"2021-11-27T12:45:32","slug":"what-is-the-equation-of-normal-to-parabola","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-equation-of-normal-to-parabola\/","title":{"rendered":"Equation of Normal to Parabola in all Forms"},"content":{"rendered":"

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

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

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

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

\n

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

Example : <\/span> Find the normal to the parabola \\(y^2 = 32x\\) at (3, 1).<\/p>\n

Solution : <\/span>We have, \\(y^2 = 32x\\)

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

\n\\(\\implies\\) a = 8

\nHence, required equation of normal is y – 1 = \\(-1\\over 16\\)(x – 3)

\n=> 16y + x = 19

<\/p>\n\n\n

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

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

\n

y = mx – 2am – a\\(m^3\\)<\/p>\n

The foot of the normal is (\\(am^2\\), -2am)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the normal to the parabola \\(y^2 = 4x\\) whose slope is 3.<\/p>\n

Solution : <\/span>We have, \\(y^2 = 4x\\)

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

\na = 1

\nHence, required equation of normal is y = 3x – 33

<\/p>\n\n\n

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

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

\n

y + tx = 2at + a\\(t^3\\)<\/p>\n<\/blockquote>\n

Note –<\/strong> Point of intersection of the normals at the points \\(t_1\\) & \\(t_2\\) is<\/p>\n

\n

[(a\\({t_1}^2 + {t_2}^2 + t_1t_2 + 2\\)), -a\\(t_1t_2(t_1 + t_2)\\)].<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> If two normals drawn from any point to the parabola \\(y^2\\) = 4ax make angle \\(\\alpha\\) and \\(\\beta\\) with the axis such that \\(tan\\alpha\\).\\(tan\\beta\\) = 2, then find the locus of this point.<\/p>\n

Solution : <\/span>Let the point is (h, k). The equation of any normal to the parabola \\(y^2\\) = 4ax is

\ny = mx – 2am – a\\(m^3\\)

\nSince it passes through (h,k)

\nk = mh – 2am – a\\(m^3\\)

\na\\(m^3\\) + m(2a – h) + k = 0 ……(i)

\n\\(m_1\\), \\(m_2\\) and \\(m_3\\) are roots of equation, then

\n\\(m_1\\).\\(m_2\\).\\(m_3\\) = \\(-k\\over a\\)

\nbut \\(m_1\\).\\(m_2\\) = 2, \\(m_3\\) = \\(-k\\over 2a\\)

\n\\(m_3\\) is the root of (i)

\n\\(\\therefore\\) \\(k^2\\) = 4ah

\nThus locus is \\(y^2\\) = 4ax.<\/p>\n\n\n

Hope you learnt equation of normal 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
Previous – Equation of Tangent to Parabola in all Forms<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

The equation of normal to parabola in point form, slope form and parametric form are given below with examples. Equation of Normal to Parabola \\(y^2 = 4ax\\) (a) Point form : The equation of normal to the given parabola at its point (\\(x_1, y_1\\)) is y – \\(y_1\\) = \\(-y_1\\over 2a\\)(x – \\(x_1\\)) Example : …<\/p>\n

Equation of Normal 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":[518,521,520,519],"yoast_head":"\nEquation of Normal to Parabola in all Forms - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn equation of normal 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\/what-is-the-equation-of-normal-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 Normal to Parabola in all Forms - 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