2005 AIME II Problem 15

Let \omega_{1} and \omega_{2} denote the circles x^{2}+y^{2}+10 x-24 y-87=0 and x^{2}+y^{2}- 10 x-24 y+153=0, respectively. Let m be the smallest positive value of a for which the line y=a x contains the center of a circle that is internally tangent to \omega_{1} and externally tangent to \omega_{2}. Given that m^{2}=p / q, where p and q are relatively prime positive integers, find p+q.