Question

AP and AQ are the two tangents drawn to a circle with centre O, these tangents are inclined to each other at an angle ${60}^0$. Find $\angle APQ$ in the given figure.

• A. ${80}^{\circ }$
• B. ${70}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$

By Theorem- If two tangents AP and AQ are drawn to a circle with centre O from an external point A, then$\angle PAQ=2\angle OPQ=2\angle OQP$
$\Rightarrow$ $\angle OPQ=\frac{1}{2}\angle PAQ$

[Given: $\angle PAQ$ = ${60}^{\circ }$]
$\therefore$ $\angle OPQ={30}^{\circ }$

and, $\angle OPA=\angle OPQ+\angle APQ$
$\Rightarrow$ $\angle APQ=\angle OPA-\angle OPQ$ ----(i)

$\because$ By Theorem- Tangent is perpendicular to the radius through the point of contact

$\therefore \angle OPA={90}^{\circ }$

On putting values of $\angle OPA$ and $\angle OPQ$ in (i), we get,
$\angle APQ={90}^{\circ }-{30}^{\circ }={60}^{\circ }$