26 -1 On the same plane, if all circles with radius $a$ pass through a fixed point, then the locus of the centers of these equal circles is
(A) a point.
(B) a straight line.
(C) two straight lines.
(D) a circle.
(E) two circles.
(11th American High School Mathematics Examination, 1960)
[Sol] The equation $(x-\alpha)^{2}+(y-\beta)^{2}=a^{2}$ represents a circle with center at $(\alpha, \beta)$ and radius $a$.

Conversely, consider $(\alpha-x)^{2}+(\beta-y)^{2}=a^{2}$, which also represents many circles with centers at $(x, y)$ and passing through $(\alpha, \beta)$.
Therefore, the locus of the center $(x, y)$ is a circle.
Hence, the answer is $(D)$.