Find the circular cone with vertex O and axis Oz through…

Find the circular cone with vertex $O$ and axis $Oz$ through $M(3,4,-5)$.

Explanation of Method

The circular cone with vertex O and axis Oz that passes through a given point.

The cone has the form $x^2+y^2=k^2z^2$, with $k=\tan\alpha$. Substituting $M$: $25=25k^2$, so $k^2=1$.

\[x^2+y^2=z^2\]

The half-angle is $45^\circ$, and the plane $z=c$ cuts the cone in a circle of radius $|c|$.

$M(3,4,-5)$ $O$

The cone $x^2+y^2=z^2$ (half-angle $45^\circ$) through $M(3,4,-5)$; the generator $OM$ is red, the circle $z=-5$ blue.

\[\boxed{\,x^2+y^2=z^2\,}\]