Problem source

Show that in polar coordinates the gradient of any curve 
at the point $(r,\theta)$ is
\[
\frac{ \ \ \dfrac{\d r }{\d\theta} \tan\theta + r \ \ }
{ \dfrac{\d r }{\d\theta} -r\tan\theta}\,.
\]
\noindent \begin{center}
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\par\end{center}
A mirror is designed so that if an incident ray of light is parallel
to a fixed line $L$ the  reflected ray passes through a fixed point $O$
on $L$. Prove that the mirror intersects any plane containing $L$ in
a parabola. You should assume that the angle between the incident
ray and 
the normal to the mirror is the same as the 
angle between the reflected ray and the normal.