I hadn't seriously attempted this problem until just now (I saw it last month and got scared ). But I just solved it without too much trouble!
Let , the field of rational functions in . Consider as elements of (which is a P.I.D.). Then it is easy to see that are relatively prime in . Hence we can find rational functions such that . Multiplying through by a common denominator, we find that there exist polynomials such that . The right side of this equation can vanish for only a finite number of possible values of , hence there are only a finite number of values of which are candidates for an intersection point of the curves and . Reversing the roles of , we arrive at the same conclusion for ; hence there are only a finite number of possible intersection points of the two curves.