1978 AMC 12 Problem 10

For $A\ne P,$ the nearest point of the circle lies on the ray from $P$ through $A$

Require that this radial nearest point be the fixed point $B$

For any $A\ne P,$ the closest point of the circle to $A$ is where the ray from $P$ through $A$ meets the circle. This point is $B$ exactly when $A$ lies on the ray from $P$ through $B.$ The center $P$ also qualifies because every point on the circle is equally distant from it. Thus the locus is that ray.

Therefore, the correct answer is B.