1952 AMC 12 Problem 8

Question

Two equal circles in the same plane cannot have the following number of common tangents:

1

2

3

4

None of these

Answer: A

Concepts:circle tangent line casework

Difficulty rating: 1430

Small hint:

Consider intersecting, externally tangent, and disjoint equal circles

Big hint:

An internal tangency would require unequal radii unless the circles coincide

Solution:

Two distinct equal circles have

2

common tangents when they intersect,

3

when externally tangent, and

4

when disjoint. They cannot be internally tangent, because internal tangency requires the center distance to equal the difference of the radii, which is

0.

At center distance

0

the equal circles coincide and have infinitely many common tangents.

Thus exactly

1

common tangent is impossible, so the correct answer is A.