We give an asymptotic formula for the number of common perpendiculars of length tending to infinity between two divergent geodesics in finite volume real hyperbolic manifolds, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular curve recovering results of Sarnak, and to prove a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields. This is a joint work with Jouni Parkkonen.