Two mirrors are placed in the first quadrant of the xy-plane (perpendicular to the plane). The first mirror along the line y = b-x (for some b > 0) and facing the point (0,0). The second mirror along the line y = mx (where m > 0) and facing the point (b,0). A light source at point (a,0), 0 < a < b, shoots a beam of light into the first quadrant parallel to the first mirror. Find m such that when the beam is reflected exactly once by each mirror, it passes through the original light source at point (a,0).

Experiment with GSP gives the following table, of b/a vs. m:

b/a     m
 2      3
 3      5
 4      7
 5      9
 6     11

which seems to indicate m = 2b/a - 1.

As to a proof, I don't know.

Posted by Charlie on 2006-10-08 13:04:40