Let \(w_1\) and \(w_2\) denote the circles \(x^2+y^2+10x-24y-87=0\) and \(x^2 +y^2-10x-24y+153=0,\) respectively. Let \(m\) be the smallest positive value of \(a\) for which the line \(y=ax\) contains the center of a circle that is externally tangent to \(w_2\) and internally tangent to \(w_1.\) Given that \(m^2=\frac pq,\) where \(p\) and \(q\) are relatively prime integers, find \(p+q.\)

(第二十三届AIME2 2005 第15题)