2013 AMC 10B Problem 14

Question

Define

a\clubsuit b=a^2b-ab^2 .

Which of the following describes the set of points

(x, y)

for which

x\clubsuit y=y\clubsuit x ?

A finite set of points

One line

Two parallel lines

Two intersecting lines

Three lines

Expand both custom-operation expressions

Factor the resulting equation into line factors

The condition is

x\clubsuit y=y\clubsuit x

, so

x^2y-xy^2=y^2x-yx^2

.

Combining like terms gives

2x^2y-2xy^2=2xy(x-y)=0

.

Thus

x=0

,

y=0

, or

x=y

. These are three lines.

Thus, the correct answer is E .

Problem 14 in other years