Answer to Question #294581 in Math for Amber

Question

Two particles move in concentric circles, of center O and radii a, b (a < b), with uniform speeds u, v in the same sense. Let P, Q be the positions of the particles at a moment when their relative velocity is along the line joining them. 

 

(a) Obtain expressions for the ratios b : a and v : u in terms of α and β, where α = ∠OQP 

and β = ∠OPQ.

(b) Show further that if θ = ∠POQ, 

cos θ = (au + bv) / (bu+av)