Printable

2006 AMC 10A Problem 11

Problem only Back to the problem

2006 AMC 10A Problem 11AMC 10 Step by Step

Name: Date: September 13, 2026

Coordinate Geometry · difficulty 2 of 5 · about 1 min · position P11-15

Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$ ?

A)
$\text{the empty set}$
B)
$\textrm{one point}$
C)
$\textrm{two lines}$
D)
$\textrm{a circle}$
E)
$\textrm{the entire plane}$

Answer

Key insight

Expanding leaves xy = 0, which holds exactly when x = 0 or y = 0: the two coordinate axes.

Solution

Expand the left side: $(x + y)^2 = x^2 + 2xy + y^2$ . Cancelling $x^2 + y^2$ from both sides leaves
$2xy = 0 \quad\Longrightarrow\quad xy = 0.$
A product is zero exactly when a factor is zero, so the solutions are all points with $x = 0$ (the $y$ -axis) together with all points with $y = 0$ (the $x$ -axis). That is two lines crossing at the origin.

The answer is $\boxed{\textbf{(C)}\ \text{two lines}}$ .

Why this works

The graph of an equation is its full solution set, so simplify the equation as far as possible and then describe every solution. A factored equation $f \cdot g = 0$ always graphs as the union of the graphs of $f = 0$ and $g = 0$ ; here each factor is a coordinate axis.

The trap

Reading xy = 0 as the single point (0, 0), or as one line, instead of the union of two lines.

Common mistakes

Reading $xy = 0$ as the single point $(0, 0)$ , or as one line, instead of the union of two lines.
Believing $(x + y)^2 = x^2 + y^2$ is an identity (the "freshman's dream") and answering the entire plane.

Techniques

Apply an identity: SFFT, sum of squares, difference of cubes, Vieta