For integers q ≥ 1, s ≥ 3 and a with gcd (a, q) = 1 and a real U ≥ 0, we obtain an asymptotic formula for the number of integer points (u1, ..., us) ∈ [1, U]s on the s-dimensional modular hyperbola u1 ⋯ us ≡ a (mod q) with the additional property gcd (u1, ..., us) = 1. Such points have a geometric interpretation as points on the modular hyperbola which are "visible" from the origin. This formula complements earlier results of the first author for the case s = 2 and a = 1. Moreover, we prove stronger results for smaller U on "average" over all a. The proofs are based on the Burgess bound for short character sums.