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My reasoning for part A follows as such: Because the radius of curvature is smaller in the second mirror than in the initial elliptical mirror, every point on the second mirror (other than the tangent point) is now closer to A and B. Thus, any light ray originating from A or B and heading towards any of these points will reach the second mirror in less time than in the case of the elliptical mirror. Thus, the tangent point must be the point of maximum time since it is now the furthest point on the second mirror from either A or B. However, I am not really sure about this answer.

As far as part B, I can't seem to wrap my head around it. I can't quite figure out what the question means when it asks about ray paths exhibiting a point of inflection. How can rays exhibit a point of inflection?