2019 AMC 12B Problem 6

Question

In a given plane, points

A

and

B

are

10

units apart. How many points

C

are there in the plane such that the perimeter of

\triangle ABC

is

50

units and the area of

\triangle ABC

is

100

square units?

0

2

4

8

infinitely many

With

AB

fixed, the perimeter condition puts

C

on an ellipse with foci

A

and

B

Area

100

with base

10

needs height

20;

compare that to the ellipse’s greatest height

The perimeter condition gives

CA+CB=50-10=40,

so

C

lies on an ellipse with foci

A,B

and major axis

2a=40.

Thus

a=20

and

c=5,

so the semi-minor axis is

b=\sqrt{a^2-c^2}=\sqrt{375}\approx19.36.

For area

100

with base

AB=10,

the height from

C

must be

\dfrac{2\cdot100}{10}=20.

But the greatest possible height on the ellipse is

b\approx19.36\lt20,

so no such

C

exists.

Thus, A is the correct answer.

Problem 6 in other years