On the number of distances between the coordinates of points on modular hyperbolas

For a prime p > 2, an integer a with gcd (a, p) = 1 and 1 ≤ X, Y < p we give an asymptotic formula for the number of different Euclidean distances | x - y | defined by the points on the modular hyperbola {(x, y) : x y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. Furthermore, in the case X = Y = p - 1 we determine the exact number of different distances.