# Equation of Tangent to a Circle – Condition of Tangency

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  to a circle and equation of tangent to a circle with example.

## Condition of Tangency :

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 = r

## Equation of Tangent to a Circle Formula

(i)  The Tangent at a point ($$x_1,y_1$$) on the circle $$x^2$$ + $$y^2$$ = $$a^2$$ is

$$xx_1 + yy_1$$ = $$a^2$$

(ii) The Tangent at the point (acost, asint) on the circle $$x^2$$ + $$y^2$$ = $$a^2$$ is

xcost + ysint = a

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

(iii) The equation of 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

(iv)  If line y = mx + c is a straight line touching the circle $$x^2$$ + $$y^2$$ = $$a^2$$, then

c = $$\pm a\sqrt{1 + m^2}$$

and contact points are

($$\mp am\over \sqrt{1 + m^2}$$, $$\pm a\over sqrt{1 + m^2}$$)

and the equation of tangent is

y = mx $$\pm a\sqrt{1 + m^2}$$

(iv) The equation of tangent with slope m of the circle $$(x-h)^2 + (y-k)^2$$ = $$a^2$$ is

(y – k) = m(x – h) $$\pm a\sqrt{1 + m^2}$$

Example : Find the tangent to the circle $$x^2 + y^2 – 2ax$$ = 0 at the point (5, 6).

Solution : 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.

5x + 6y – a(x + 5) = 0

$$\implies$$ 5x + 6y – ax – 5a = 0
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