The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
$\textbf{(A) }\text{a point}\qquad \textbf{(B) }\text{a straight line}\qquad \textbf{(C) }\text{two straight lines}\qquad \textbf{(D) }\text{a circle}\qquad  \textbf{(E) }\text{two circles}$
[asy] draw(circle((0,0),50)); dot((0,0)); dot((-30,-40)); draw(circle((-30,-40),50),dotted); dot((50,0)); draw(circle((50,0),50),dotted); draw((-30,-40)--(0,0)--(50,0)); [/asy]
If a circle passes through a point, then the point is $a$ units away from the center.  That means that all of the centers are $a$ units from the point.  By definition, the resulting figure is a circle, so the answer is $\boxed{\textbf{(D)}}$.