Question 15
Circles $C_1$ and $C_2$ intersect at two points, one of which is $(9, 6),$ and the product of their radii is $68.$ The $x$-axis and the line $y = mx,$ where $m \gt 0,$ are tangent to both circles. It is given that $m$ can be written in the form $\frac{a\sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a + b + c.$