G1.4 In Figure 1, $P A$ touches the circle with centre $O$ at $A$. If $P A=6, B C=9, P B=d$, find the value of $d$.
It is easy to show that $\triangle P A B \sim \triangle P C A$
$$
\begin{array}{l}
\frac{P A}{P B}=\frac{P C}{P A} \quad \text { (ratio of sides, } \sim \Delta \text { 's) } \\
\frac{6}{d}=\frac{9+d}{6} \\
36=9 d+d^{2} \\
d^{2}+9 d-36=0 \\
(d-3)(d+12)=0 \\
d=3 \text { or }-12 \text { (rejected) }
\end{array}
$$