{"id":3426,"date":"2021-07-24T13:14:37","date_gmt":"2021-07-24T13:14:37","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3426"},"modified":"2021-11-20T22:58:24","modified_gmt":"2021-11-20T17:28:24","slug":"equation-of-tangent-to-a-circle","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-tangent-to-a-circle\/","title":{"rendered":"Equation of Tangent to a Circle – Condition of Tangency"},"content":{"rendered":"

When a straight line meet a circle on two coincident points then it is called the tangent to a circle. Here, you will learn condition of line to be a tangent\u00a0 to a circle and equation of tangent to a circle with example.<\/p>\n

Condition of Tangency :<\/h2>\n

The line L = 0 touches the circle S = 0 if P the length of the perpendicular from the center to that line and radius of the circle r are equal i.e.<\/p>\n

\n

P = r<\/strong><\/p>\n<\/blockquote>\n

Equation of Tangent to a Circle Formula<\/h2>\n

(i)\u00a0 The Tangent at a point (\\(x_1,y_1\\)) on the circle \\(x^2\\) + \\(y^2\\) = \\(a^2\\) is<\/p>\n

\n

\\(xx_1 + yy_1\\) = \\(a^2\\)<\/strong><\/p>\n<\/blockquote>\n

(ii) The Tangent at the point (acost, asint) on the circle \\(x^2\\) + \\(y^2\\) = \\(a^2\\) is<\/p>\n

\n

xcost + ysint = a<\/strong><\/p>\n<\/blockquote>\n

The point of intersection of the tangents at the points \\(P(\\alpha)\\) and \\(Q(\\beta)\\) is (\\(acos{{\\alpha + \\beta}\\over 2}\\over cos{{\\alpha – \\beta}\\over 2}\\), \\(asin{{\\alpha + \\beta}\\over 2}\\over cos{{\\alpha – \\beta}\\over 2}\\)).<\/p>\n

(iii) The equation of tangent at the points (\\(x_1,y_1\\)) on the circle \\(x^2 + y^2 + 2gx + 2fy + c\\) = 0 is<\/p>\n

\n

\\(xx_1\\) + \\(yy_1\\) + \\(g(x + x_1)\\) + \\(f(y + y_1)\\) + c = 0<\/strong><\/p>\n<\/blockquote>\n

(iv)\u00a0 If line y = mx + c is a straight line touching the circle \\(x^2\\) + \\(y^2\\) = \\(a^2\\), then<\/p>\n

\n

c = \\(\\pm a\\sqrt{1 + m^2}\\)<\/strong><\/p>\n<\/blockquote>\n

and contact points are<\/p>\n

\n

(\\(\\mp am\\over \\sqrt{1 + m^2}\\), \\(\\pm a\\over sqrt{1 + m^2}\\))<\/strong><\/p>\n<\/blockquote>\n

and the equation of tangent is<\/p>\n

\n

y = mx \\(\\pm a\\sqrt{1 + m^2}\\)<\/strong><\/p>\n<\/blockquote>\n

(iv) The equation of tangent with slope m of the circle \\((x-h)^2 + (y-k)^2\\) = \\(a^2\\) is<\/p>\n

\n

(y – k) = m(x – h) <\/strong>\\(\\pm a\\sqrt{1 + m^2}\\)<\/strong><\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the tangent to the circle \\(x^2 + y^2 – 2ax\\) = 0 at the point (5, 6).<\/p>\n

Solution : <\/span>Since the tangent at the points (\\(x_1,y_1\\)) on the circle \\(x^2 + y^2 + 2gx + 2fy + c\\) = 0 is \\(xx_1\\) + \\(yy_1\\) + \\(g(x + x_1)\\) + \\(f(y + y_1)\\) + c = 0.

\n5x + 6y – a(x + 5) = 0

\n\\(\\implies\\) 5x + 6y – ax – 5a = 0
br>\n<\/p>\n\n\n\n

\n
Next – Equation of Normal to a Circle with Examples<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Diameter Form of Circle – Equation and Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

When a straight line meet a circle on two coincident points then it is called the tangent to a circle. Here, you will learn condition of line to be a tangent\u00a0 to a circle and equation of tangent to a circle with example. Condition of Tangency : The line L = 0 touches the circle …<\/p>\n

Equation of Tangent to a Circle – Condition of Tangency<\/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":[23],"tags":[181,189,190],"yoast_head":"\nEquation of Tangent to a Circle - Condition of Tangency<\/title>\n<meta name=\"description\" content=\"In this post, you will learn condition of tangency to circle and how to find equation of tangent to a circle 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-a-circle\/\" \/>\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 a Circle - 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