1960 AHSME Problems/Problem 12

Problem 12

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

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

Solution

[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 $\boxed{\textbf{(D)}}$.

See Also

1960 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
All AHSME Problems and Solutions