2023 AMC 10A Problem 22

Question

Circle

C_1

and

C_2

each have radius

1,

and the distance between their centers is

\frac{1}{2}.

Circle

C_3

is the largest circle internally tangent to both

C_1

and

C_2.

Circle

C_4

is internally tangent to both

C_1

and

C_2

and externally tangent to

C_3.

What is the radius of

C_4?

\dfrac{1}{14}

\dfrac{1}{12}

\dfrac{1}{10}

\dfrac{3}{28}

\dfrac{1}{9}

Answer: D

Concepts:tangent circles coordinate geometry symmetry

Difficulty rating: 2270

Small hint:

Put the centers of

C_1, C_2

at

\left(\pm\frac14, 0\right);

C_3

is centered at the origin with radius from

1 - r_3 = \frac14

Big hint:

Let

C_4

be centered at

(0, y)

with radius

r;

internal tangency to

C_1

and external tangency to

C_3

give two equations

Solution:

Put the centers of

C_1, C_2

at

\left(\pm\frac14, 0\right).

By symmetry the largest circle inside both sits at the origin with radius

r_3,

where

1 - r_3 = \frac14,

so

r_3 = \frac34.

Let

C_4

be centered at

(0, y)

with radius

r.

Internal tangency to

C_1

gives

\sqrt{\frac1{16} + y^2} = 1 - r,

and external tangency to

C_3

gives

y = \frac34 + r.

Substitute the second into the first:

\frac1{16} + \left(\frac34 + r\right)^2 = (1 - r)^2.

This collapses to

\frac72 r = \frac38,

so

r = \frac{3}{28}.

Therefore, the answer is D.

2023 AMC 10A Problem 22

Answer: D

Difficulty rating: 2270

Small hint:

Big hint:

Solution:

Problem 22 in other years