Question

Two tangents PA and PB are drawn from an external point P to the circle with centre O, such that $\angle$APB =${120}^{\circ }$what is the relation between OP and AP?

• A. OP = $\frac{1}{2}$ AP
• B. AP = $\frac{2}{3}$ OP
• C. OP = 2 AP
• D. OP = AP

Answer 1

The correct option is C

OP = 2 AP

Given that $\angle APB={120}^{\circ }$

Also, we know that if two tangents are drawn from an external point to a circle, then the line joining the external point and the centre of the circle bisects the angle between the tangents.

$⟹\angle APO=\angle OPB={60}^{\circ }$

Thus, cos$\angle OPA=cos{60}^{\circ }=\frac{AP}{OP}$
$⟹\frac{1}{2}$ = $\frac{AP}{OP}$

Thus, $OP=2AP$