9. (15 points) As shown in Figure 2, circle $\odot M$ is externally tangent to circle $\odot O$ at point $C$, with radii $r$ and $R$ for $\odot M$ and $\odot O$ respectively. Line $TPQ$ is tangent to $\odot M$ at point $T$, and intersects $\odot O$ at points $P$ and $Q$. Find the value of $\frac{CQ - CP}{PQ}$.
II. 9. As shown in Figure 6, extend $PC$ and $QC$ to intersect $\odot M$ at points $D$ and $E$ respectively.

Since $M, C, O$ are collinear,
$\Rightarrow$ Isosceles $\triangle MCD \sim$ Isosceles $\triangle OCP$
$\Rightarrow \frac{CD}{CP}=\frac{r}{R} \Rightarrow \frac{PD}{CP}=\frac{R+r}{R}$
$\Rightarrow \frac{PD \cdot PC}{PC^2}=\frac{R+r}{R}$.
Also, $PD \cdot PC=PT^2$, hence
$\frac{PT^2}{PC^2}=\frac{R+r}{R}$
$\Rightarrow \frac{PT}{PC}=\sqrt{\frac{R+r}{R}}$.
Similarly, $\frac{QT}{QC}=\sqrt{\frac{R+r}{R}}$.
From equations (1) and (2),
$$
\begin{array}{l}
\frac{QT-PT}{QC-PC}=\sqrt{\frac{R+r}{R}} \Rightarrow \frac{PQ}{QC-PC}=\sqrt{\frac{R+r}{R}} \\
\Rightarrow \frac{QC-PC}{PQ}=\sqrt{\frac{R}{R+r}} .
\end{array}
$$