Question

A tangent is drawn to a circle of radius r from an external point P .If length of tangent from a point to the circle is twice the minimum distance between circle and the point, what is the length of the tangent?

• A. $\frac{3}{2r}$
• B. 3 r
• C. $\frac{2r}{3}$
• D. $\frac{4r}{3}$

Answer 1

The correct option is D

$\frac{4r}{3}$

Lets draw tje corc;e, the external point P and the tangents as per given conditions.

Here we can see that PR is the minimum distance between P and circle. Let it be x. its then given that PQ

=$2x$

$△OPQ$ is a right angled triangle

$i.e.,(x+r{)}^2=r^2+4x^2$

$i.e.,x^2+r^2+2rx=r^2+4x^2$

$3x^2-2rx=0$

$x(x-\frac{2r}{3})=0$

here $x=0$ means P is on the circle itself. But its given that P is outside the circle.

$\therefore x-\frac{2r}{3}=0$

$\Rightarrow x=\frac{2r}{3}$

$\therefore$ Length of tangent =2x

$=\frac{4r}{3}$