I'm not sure what Uli was planning on doing with this problem, but here's the resolution.
Because the position of the line depends upon two variables, it's not enough to say they are both uniformly distributed. We need to know how these two variables interact in order to give the right answer.
As you might expect, what is uniform in one coordinate system is not uniform in another coordinate system. Thus, the two answers given are really the answer to two similar but different problems.
