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So let's say you have a curved mirror, $y=y(x)$ with this property: Whenever a ray of light emanates from the origin, it reflects parallel to the x axis.

Find the equation of the mirror.

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    $\begingroup$ Is this homework or a puzzle you already know the answer to? $\endgroup$ – Qiaochu Yuan Oct 13 '10 at 22:46
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    $\begingroup$ en.wikipedia.org/wiki/Parabolic_reflector $\endgroup$ – Aryabhata Oct 13 '10 at 22:47
  • $\begingroup$ How much mathematical background do you have? Are you comfortable with solutions involving calculus and vector algebra? $\endgroup$ – Justin L. Oct 14 '10 at 1:15
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This sounds like homework. Can you translate the words into an equation? Think about what happens when the light ray strikes the mirror.

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Making Ross's answer more explicit: remember that in the vicinity of the point where the light ray from the origin strikes your mirror, you can replace the curved mirror with the corresponding tangent line. Try deriving a differential equation for your $f(x)$ (hint: you may have to use the formula for the tangent of the difference of two angles) such that the reflected ray has zero slope (i.e., is horizontal).

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