Question

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

the line segment from $P$ to $B$

the ray beginning at $P$ and passing through $B$

a ray beginning at $B$

a circle whose center is $P$

a circle whose center is $B$

Answer: B

Concepts:circle extremal argument

Difficulty rating: 1510

Small Hint:

For $A\ne P,$ the nearest point of the circle lies on the ray from $P$ through $A$

Big Hint:

Require that this radial nearest point be the fixed point $B$

Solution:

For any $A\ne P,$ the closest point of the circle to $A$ is where the ray from $P$ through $A$ meets the circle. This point is $B$ exactly when $A$ lies on the ray from $P$ through $B.$ The center $P$ also qualifies because every point on the circle is equally distant from it. Thus the locus is that ray.

Therefore, the correct answer is B.

1978 AMC 12 Problem 10

Answer: B

Difficulty rating: 1510

Small Hint:

Big Hint:

Solution:

Problem 10 in Other Years