Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 5A / c

The signed displacement of the sphere centre from the plane is $g\cdot n-p$. Orthogonal projection gives the circle centrePythagoras givesA real circle exists when $|p-g\cdot n|\le A$: equality gives tangency and radius zero, while strict inequality gives a genuine circle.