# 1960 AHSME Problems/Problem 12

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## 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 (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 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
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