2. Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ have radii 3 and 7 respectively. The circles intersect at distinct points $A$ and $B$. A point $P$ outside $\mathcal{C}_{2}$ lies on the line determined by $A$ and $B$ at a distance of 5 from the center of $\mathcal{C}_{1}$. Point $Q$ is chosen on $\mathcal{C}_{2}$ so that $P Q$ is tangent to $\mathcal{C}_{2}$ at $Q$. Find the length of the segment $P Q$.
2. Ans: 4
$P$ lies on the radical axis of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ and hence has equal power with respect to both circles. Now
$$
\begin{aligned}
P Q^{2} & =\text { Power of } P \text { with respect to } \mathcal{C}_{2} \\
& =\text { Power of } P \text { with respect to } \mathcal{C}_{1} \\
& =5^{2}-3^{2}=16 .
\end{aligned}
$$