However, each solution $ (x, y) $ corresponds to a unique pair $ (a, b) $, and since $ x $ and $ y $ are determined uniquely, and the hyperbola is symmetric, we must check for duplicates. But since $ (a, b) $ and $ (b, a) $ would give different $ x $ and $ y $, and both are included, all 30 pairs produce valid, distinct lattice points. - Abu Waleed Tea