1978 AHSME Problems/Problem 10

Problem 10

If $\mathit{B}$ is a point on circle $\mathit{C}$ with center $\mathit{P}$ , then the set of all points $\mathit{A}$ in the plane of circle $\mathit{C}$ such that the distance between $\mathit{A}$ and $\mathit{B}$ is less than or equal to the distance between $\mathit{A}$ and any other point on circle $\mathit{C}$ is

$\textbf{(A) }\text{the line segment from }P \text{ to }B\qquad\\ \textbf{(B) }\text{the ray beginning at }P \text{ and passing through }B\qquad\\ \textbf{(C) }\text{a ray beginning at }B\qquad\\ \textbf{(D) }\text{a circle whose center is }P\qquad\\ \textbf{(E) }\text{a circle whose center is }B$

Solution

Begin by drawing circle P and point B. To satisfy the conditions of the problem, A needs to be in a position where it is closer to B. This can only happen if A and B are on the same line, so we choose from answer choices A and B.

We can pick some arbitrary point A outside circle P that is collinear with B and see that the conditions still hold, so the answer is $\boxed{\textbf{(B)}}.$

1978 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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1978 AHSME Problems/Problem 10

Problem 10

Solution

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