2007 AMC 10B Problem 10

Question

Two points

B

and

C

are in a plane. Let

S

be the set of all points

A

in the plane for which

\triangle ABC

has area

1.

Which of the following describes

S?

two parallel lines

a parabola

a circle

a line segment

two points

Fix

BC

as the base; the area determines the height from

A

All points at a fixed distance from line

BC

lie on two parallel lines

Taking

BC

as the base, the area is

\dfrac12(BC)d,

where

d

is the distance from

A

to line

BC.

The area equals

1

exactly when

d=\dfrac{2}{BC}.

The points at this fixed distance from line

BC

form two lines parallel to

BC,

one on each side.

Thus, the correct answer is A.

Problem 10 in other years