2019 AMC 12B Problem 20

Problem 20 of 25HarderGeometry

Points

A(6,13)

and

B(12,11)

lie on circle

\omega

in the plane. Suppose that the tangent lines to

\omega

at

A

and

B

intersect at a point on the

x

-axis. What is the area of

\omega?

Show solution

Solution

Let

P=(x,0)

be the intersection. Equal tangent lengths give

PA=PB,

so

(x-6)^2+13^2

=(x-12)^2+11^2,

yielding

x=5

and

P=(5,0).

The center

O=(h,k)

satisfies

OA\perp PA

and

OB\perp PB.

With

PA=(1,13)

and

PB=(7,11),

these give

h+13k=175

and

7h+11k=205,

so

O=\left(\dfrac{37}{4},\dfrac{51}{4}\right).

Then

r^2=OA^2

=\left(\dfrac{13}{4}\right)^2+\left(\dfrac14\right)^2

=\dfrac{170}{16}=\dfrac{85}{8},

so the area is

\dfrac{85\pi}{8}.

Thus, C is the correct answer.

More practice

Concepts: circle · tangent line · coordinate geometry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.