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{\bf Question}

Verify that the equation $$r = \frac{a}{\sin^2
\frac{1}{2}\theta}$$ is the polar equation of a parabola.  Prove
that $\ds \phi = \pi - \frac{1}{2} \theta$ and use this to deduce
the parabolic mirror property.

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{\bf Answer}

$\ds r = \frac{a}{\sin^2\frac{1}{2}\theta}={2a}{1-\cos \theta}$

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So $\ds \frac{2a}{r} = 1 - \cos \theta.$  This is the standard
equation of a conic with eccentricity 1.  i.e. a parabola
\begin{eqnarray*} \cot \phi & = & \frac{1}{r} \frac{dr}{d\theta}
\\ & = & \frac{1}{r} a \left(-2\left(\sin\frac{1}{2}
\theta\right)^{-3}\right)\frac{1}{2} \cos \frac{1}{2} \theta \\ &
= & -\cot \frac{1}{2} \theta \\ {\rm So \ \ \ } \phi & = & \pi -
\frac{1}{2} \theta \end{eqnarray*}

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