A ray of light starts at the point A(2,3). It first strikes the…
Question
A ray of light starts at the point A(2,3). It first strikes the x-axis, reflects off it, then strikes the y-axis, reflects off it, and finally passes through the point B(7,5). All reflections obey the law of reflection (angle of incidence equals angle of reflection). The minimum possible total length of the ray's path from A to B equals sqrt(N) for some positive integer N. Find N.
Step-by-step solution
To minimize a reflected path, unfold the reflections.
The ray hits the x-axis first, then the y-axis.
Reflect the START point A successively across the two mirrors in the order they are met.
Reflect A(2,3) across the x-axis: A'(2,-3).
Reflect A' across the y-axis: A''(-2,-3).
The minimum total path length equals the straight-line distance from the doubly-reflected source A'' to the destination B, because the unfolded path is a straight segment.
Final answer145
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