Suppose a parabolic mirror is cut with the Cartesian plane so that we get a parabola with the equation y = x^2. On a randomly selected point of the graph - M(x_0, y_0) - a light beam is falling, perpendicularly to y-axis. Find coordinates of the point where this beam crosses y-axis (after reflecting from the mirror) and prove that the distance between this point and vertex of the given graph is a constant.

I had this in a test at school, not the most difficult exercise I'd say but not a piece of cake either.

P.S. I translated it directly from Latvian, so if anything is unclear, then don't hesitate asking me. I'll try to be as elaborate as I can.

Originally posted by kbaumen For all you calculus fans out there:

Suppose a parabolic mirror is cut with the Cartesian plane so that we get a parabola with the equation y = x^2. On a randomly selected point of the graph - M(x_0, y_0) - a light beam is falling, perpendicularly to y-axis (x-axis). Find coordinates of the point where this beam crosses y-axis (after reflecting fro ...[text shortened]... if anything is unclear, then don't hesitate asking me. I'll try to be as elaborate as I can.

This can be done with straight-up Euclidean geometry, although I'm not giving away how yet; with your typo amended as italicised, the beam will cross the y-axis at the point (0, 1/4), so that the distance from this point and the vertex is the constant 1/4.