Question 58
Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0,0,0)$, and let $P$ be a point in space with coordinates $(3,4,8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.
Supplemental figure: Vertical section through the cone axis and vertex after horizontal rotation.
Supplemental figure: Vertical section through the cone axis and vertex after horizontal rotation.