10. (20 points) Given that circles $C_{1}$ and $C_{2}$ intersect at two points, one of which has coordinates $(9,6)$, and the product of the radii of the two circles is 68. If the x-axis and the line $y=m x (m>0)$ are both tangent to the two circles, find the value of $m$.


10. Let $r_{1}, r_{2}$ be the radii of circles $C_{1}, C_{2}$, and $A_{1}, A_{2}$ be their centers, $P(u, v)$ be the intersection point of the two circles.

Let $A_{1}\left(x_{1}, k x_{1}\right), A_{2}\left(x_{2}, k x_{2}\right)$, where $k$ is the tangent of the angle formed by the positive x-axis and the ray $O A_{1}$.
Note that, $r_{1}=k x_{1}, r_{2}=k x_{2}, m=\frac{2 k}{1-k^{2}}$.
From $P A_{1}^{2}=\left(k x_{1}\right)^{2}$
$$
\begin{array}{l}
\Rightarrow\left(u-x_{1}\right)^{2}+\left(v-k x_{1}\right)^{2}=k^{2} x_{1}^{2} \\
\Rightarrow x_{1}^{2}-2(u+k v) x_{1}+u^{2}+v^{2}=0 .
\end{array}
$$

Similarly, $x_{2}^{2}-2(u+k v) x_{2}+u^{2}+v^{2}=0$.
Then $x_{1} x_{2}=u^{2}+v^{2}$
$$
\begin{array}{l}
\Rightarrow r_{1} r_{2}=k^{2} x_{1} x_{2}=k^{2}\left(u^{2}+v^{2}\right) \\
\Rightarrow k=\sqrt{\frac{r_{1} r_{2}}{u^{2}+v^{2}}} \\
\Rightarrow m=\frac{2 k}{1-k^{2}}=\frac{2 \sqrt{r_{1} r_{2}\left(u^{2}+v^{2}\right)}}{u^{2}+v^{2}-r_{1} r_{2}} .
\end{array}
$$

When $u=9, v=6, r_{1} r_{2}=68$, we have
$$
m=\frac{12 \sqrt{221}}{49} .
$$