Let C be a right circular cone with apex A. Let P_1, P_2, P_3, P_4, and P_5 be points placed evenly along the circular base in that order, so that P_1P_2P_3P_4P_5 is a regular pentagon. Suppose that the shortest path from P_1 to P_3 along the curved surface of the cone passes through the midpoint of AP_2. Let h be the height of C, and r be the radius of the circular base of C. If (\frac{h}{r})^2 can be written in simplest form as \frac{a}{b}, find a + b.