Past exam of the mathematical methods I course of the University of Oxford / 2014 / 10 / b / ii / Solution

First assume $a\ne 0$. The first equation is a plane, and the second is a sphere centred at $c$. Define its signed normal separation and projected centre byThe intersection of a plane and a sphere isThe circle lies in the given plane. Its radius follows from the Pythagorean theorem, splitting $x-c=(x-q)+(q-c)$ into orthogonal components. If $a=0$, the first equation is the identity $0=0$, so the locus is the entire sphere, rather than a planar section.