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Two Mirrors (Posted on 2006-10-08) Difficulty: 2 of 5
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).

See The Solution Submitted by Bractals    
Rating: 2.5000 (4 votes)

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Some Thoughts To start Comment 4 of 4 |

Bractals was nice enough to solve my Bouncy Bouncy Revisited with a nice formula for bouncing a laser beam of slope a off of a mirror of slope b.  a'=(b(ab+2)-a)/(b(2a-b)+1))

The original path of the beam has slope -1 andbounces off a mirror of slope m so we have (after simplifying)

a'= (m^2 - 2m - 1)/(m^2 + 2m - 1)

Bouncing this off the mirror of slope -1 gives

a"= 1/a' = (m^2 + 2m - 1)/(m^2 - 2m - 1)

So from here its just a matter of finding the points and getting them to align.

  Posted by Jer on 2006-10-10 11:36:40
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