Critique of solution to falling body problem

We use some standard problem-posing question openers to pose some extension problems:

In general, what if the point is 1/3 of the way up a ladder of length L feet?

What if the point on the ladder were not 1/3 of the way up?

Under what conditions, if any, would the elliptical path be circular?

What if the wall and the ground were not perpendicular?

What if we did not use rectangular coordinates? What if we used polar coordinates or parametric equations?