6. The line segments $P Q R S$ and $W X Y S$ intersect circle $C_{1}$ at points $P, Q, W$ and $X$.

The line segments intersect circle $C_{2}$ at points $Q, R, X$ and $Y$. The lengths $Q R, R S$ and $X Y$ are 7, 9 and 18 respectively. The length $W X$ is six times the length $Y S$.
What is the sum of the lengths of $P S$ and $W S$ ?
Solution
150

Use the intersecting chords theorem on each circle. Let $a$ be the length of $S Y$. Then, in circle $Q X Y R, a(a+18)=9(9+7)$ giving solutions of $a=6,-24$. Since $a>0$ we conclude $a=6$. In circle $P W X Q, 24(24+6 \times 6)=16(16+z)$. The solution is $z=74$.
Therefore $P S+W S=74+7+9+36+18+6=150$.