1976 AMC 12 Problem 24

Question

In the adjoining figure, circle

K

has diameter

AB;

circle

L

is tangent to circle

K

and to

AB

at the center of circle

K;

and circle

M

is tangent to circle

K,

to circle

L

and to

AB.

The ratio of the area of circle

K

to the area of circle

M

is

12

14

16

18

not an integer

Answer: C

Concepts:tangent circles Pythagorean Theorem area ratio

Difficulty rating: 2200

Small hint:

Scale the radius of

K

to

2,

so circle

L

has radius

1

Big hint:

If

M

has radius

t

and center

(x,t),

use its tangencies to both larger circles

Solution:

Take

K

to have center

(0,0)

and radius

2,

with

AB

the

x

-axis. Then

L

has center

(0,1)

and radius

1.

If

M

has center

(x,t)

and radius

t,

internal tangency to

K

and external tangency to

L

give

\begin{aligned} x^2+t^2&=(2-t)^2,\\ x^2+(t-1)^2&=(1+t)^2. \end{aligned}

These simplify to

x^2=4-4t

and

x^2=4t,

so

t=\frac12.

The radius ratio is

2:\frac{1}{2}=4:1,

hence the area ratio is

16.

Therefore, the correct answer is C.

1976 AMC 12 Problem 24

Answer: C

Difficulty rating: 2200

Small hint:

Big hint:

Solution:

Problem 24 in other years