{"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
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
\nP = 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
\nxcost + 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
\nc = \\(\\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
\ny = 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