1960 AHSME Problem 12

The locus of the centers of all circles of given radius a, in the same plane, passing through a fixed point, is:

\textbf{(A) }\text{a point}\qquad \textbf{(B) }\text{ a straight line}\qquad \textbf{(C) }\text{two straight lines}\qquad \textbf{(D) }\text{a circle}\qquad \textbf{(E) }\text{two circles}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.