Question

If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = _______.

Answer 1

PA and PB are tangents drawn from P to circle with centre O.

∠APB = 60º and OA = OB = a (Radius of the circle)

We know that, the two tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to the point.

∴ ∠APO = ∠BPO = $\frac{60°}{2}$ = 30º

Also, ∠OAP = 90º (Radius is perpendicular to the tangent at the point of contact)

In right ∆OAP,

$$\begin{gathered} \sin 30°=\frac{\mathrm{OA}}{\mathrm{OP}} \\ \Rightarrow \frac{1}{2}=\frac{a}{\mathrm{OP}} \\ \Rightarrow \mathrm{OP}=2a \end{gathered}$$


If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = ___2a___.