Rational points in geometric progressions on certain hyperelliptic curves

We pose a simple Diophantine problem which may be expressed in the language of geometry. Let C be a hyperelliptic curve given by the equation y(2) = f(x), where f is an element of Z[x] is without multiple roots. We say that points P-i = (x(i), y(i)) is an element of C(Q) for i = 1, 2, ... , k, are in geometric progression if the numbers x(i) for i = 1, 2, ..., k, are in geometric progression. Let n >= 3 be a given integer. In this paper we show that there exist polynomials a, b is an element of Z[t] such that on the curve y(2) = a(t)x(n) + b(t) (defined over the field Q(t)) we can find four points in geometric progression. In particular this result generalizes earlier results of Berczes and Ziegler concerning the existence of geometric progressions on Pell type quadrics y(2) = ax(2) + b. We also investigate for fixed b is an element of Z, when there can exist rationals y(i), i = 1,, 4, with {y(i)(2) - b} forming a geometric progression, with particular attention to the case b = 1. Finally, we show that there exist infinitely many parabolas y(2) = ax + b which contain five points in geometric progression.