I encountered this question here (question 6) http://sections.maa.org/iowa/Activities/Contest/Problems/Probs98.htm
The Question: A bug is crawling on the coordinate plane from (7,11) to (-17, -3). The bug travels at constant speed one unit per second everywhere but quadrant II (negative x- and positive y- coordinates), where it travels at 1/2 units per second. What path should the bug take to complete its journey in minimal time?
I'm thinking that the way to solve would be to somehow dilate the quadrant II region by 2, or do some clever reflections. Then the answer would be given by a straight line path. If I try to compute an answer by calculus and Snell's law, it starts to look very very tedious.
I tried to simplify the question by placing the end point inside quadrant II, but I couldn't determine the exact path to take.
Is there an elegant way to do this problem? Thanks for the help!