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What is the equation describing the shape of a mirror with the property that light rays from a point source placed at the origin will be reflected in a beam of rays parallel to the x-axis?

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The mirror of course would be a spherical section. Further, since convex mirror converges a beam of rays to a point, and a concave mirror diverges, the mirror will be a concave one.

But how to obtain the shape of the mirror in terms of focus (a,0), say, of a parabola?

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The spherical mirror is only approximately the correct answer. The property you want is only given by a parabolic mirror.

From the Wikipedia page I can quote that the relation is $4 f y = x^2$ where $f$ is the distance of the focus to the origin of the parabola.

Your focus is at $(a, 0)$, so the equation would be $4 a x = y^2$. You can solve that equation for $y$ and have a function $y = f(x) = \sqrt{4 a x}$.

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