CAT 2025 Slot 2 QA Question 21: Two tangents drawn from a point $P$ touch a circle with center $O$ at points $Q$ and $R$. Points $A$ and $B$ lie on $PQ$ and $PR$, respectively, such that $AB$ is also a tangent to the same circle. If

Question

Two tangents drawn from a point $P$ touch a circle with center $O$ at points $Q$ and $R$. Points $A$ and $B$ lie on $PQ$ and $PR$, respectively, such that $AB$ is also a tangent to the same circle. If

Accepted Answer

The three tangent lines $PA$, $PB$ and $AB$ form $	riangle PAB$ with the given circle as an excircle opposite $P$.

For this excircle, the angle between the bisectors at $A$ and $B$ is
$\angle AOB = 90^\circ - \dfrac{1}{2}\angle APB$

$50^\circ = 90^\circ - \dfrac{1}{2}\angle APB$
$\Rightarrow \dfr