Circles \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) intersect at two points, one of which is \((9,6)\), and the product of the radii is \(68\). The x-axis and the line \(y = mx\), where \(m > 0\), are tangent to both circles. It is given that \(m\) can be written in the form \(a\sqrt {b}/c\), 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 \(a + b + c\).

(第二十届AIME2 2002 第15题)