What I did to solve (a) was to derive the formula for the lines B'A and BA' given an A'. Given that A' you know that B'=(a+h-a')
What you find is that the segments only intersect when A'=A=a in which case the two lines are identical and the locus of intersection is the entire segment.
This needs to be proven of course which would be done by finding the point of intersection and showing that it lies outside the segment endpoints.
I haven't looked at (b) in detail yet but the same method should apply.