A line through the origin slides along the circle…

A line through the origin slides along the circle $x^2+y^2=d^2,\ z=c$ ($c\neq0$). Find the surface it describes.

Explanation of Method

A line through the origin sliding along a horizontal circle sweeps a circular cone.

The line through $O$ and $(d\cos\varphi,\ d\sin\varphi,\ c)$ is $t\cdot(d\cos\varphi,\ d\sin\varphi,\ c)$. Then $t=\tfrac zc$, and

\[ x^2+y^2=\frac{d^2z^2}{c^2}. \]

\[\frac{x^2+y^2}{d^2}-\frac{z^2}{c^2}=0\]

This is a circular cone with vertex $O$, axis $Oz$ and half-angle $\alpha$, where $\tan\alpha=\tfrac{d}{|c|}$.

$O$ $x^2+y^2=d^2$

Lines through $O$ and the circle (red) form a circular cone with axis $Oz$, $\tan\alpha=\tfrac dc$. Both nappes are shown.

\[\boxed{\,\frac{x^2+y^2}{d^2}-\frac{z^2}{c^2}=0\,}\]