1991 AMC 12 Problem 22

Question

Two circles are externally tangent. Lines

PAB

and

PA'B'

are common tangents with

A

and

A'

on the smaller circle and

B

and

B'

on the larger circle. If

PA=AB=4,

then the area of the smaller circle is

1.44\pi

2\pi

2.56\pi

\sqrt8\pi

4\pi

The two circles are related by a dilation centered at

P

Use

PA=4,

PB=8

to find the dilation ratio, then apply the tangent-length relation to the smaller circle

Since

PA=4

and

AB=4,

we have

PB=8.

The homothety centered at

P

taking the smaller circle to the larger therefore has ratio

2.

If the smaller radius is

r,

its center is

3r

from

P:

the centers are on the same ray, their distances from

P

have ratio

2,

and their difference is

r+2r=3r.

Applying the right triangle formed by

P,

the small center, and

A

,

PA^2=(3r)^2-r^2=8r^2.

Thus

16=8r^2,

so

r^2=2

and the area is

2\pi.

Thus the correct answer is B.

Problem 22 in other years