= Solution
Substitute the signed displacement $\lambda=p-\mathbf n\cdot\mathbf c$ into the <Pythagorean theorem> relation from part (a):
$$
\boxed{\rho=\sqrt{a^2-(p-\mathbf n\cdot\mathbf c)^2}.}
$$
A genuine <circle> requires $|p-\mathbf n\cdot\mathbf c|<a$. Equality gives a tangent <plane> and a single point, interpreted as radius zero; a larger distance gives no intersection. Under the stipulated <circle> intersection, the radicand is positive. Its centre is $\mathbf c+(p-\mathbf n\cdot\mathbf c)\mathbf n$.
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