2023 AMC 12A Problem 18

Question

Circle

C_1

and

C_2

each have radius

1,

and the distance between their centers is

\tfrac12.

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}

By symmetry

C_3

is centered at the midpoint of

C_1C_2;

its radius is

1-\tfrac14=\tfrac34

Place

C_4

on the perpendicular bisector; its center is

1-r

from a unit circle’s center and

\tfrac34+r

from the center of

C_3

Put the centers at

O_1=\left(-\tfrac14,0\right)

and

O_2=\left(\tfrac14,0\right).

By symmetry

C_3

is centered at the origin, and internal tangency to

C_1

gives radius

1-\tfrac14=\tfrac34.

Let

C_4

have radius

r,

centered at

(0,k)

on the axis of symmetry. External tangency to

C_3

gives

k=\tfrac34+r,

and internal tangency to

C_1

gives

\sqrt{\tfrac{1}{16}+k^2}=1-r.

Substituting,

\tfrac{1}{16}+\left(\tfrac34+r\right)^2=(1-r)^2,

which simplifies to

\tfrac72 r=\tfrac38,

so

r=\dfrac{3}{28}.

Thus, the correct answer is D.

Problem 18 in other years