Question

下列哪一项描述方程 $(x + y)^2 = x^2 + y^2$ 的图像？

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

空集

the empty set

一个点

one point

两条直线

two lines

一个圆

a circle

整个平面

the entire plane

展开得 $x^2 + 2xy + y^2 = x^2 + y^2$ ，化简为 $2xy = 0$ ，也就是 $xy = 0$ 。

这恰好在 $x = 0$ 或 $y = 0$ 时成立，即两条坐标轴，所以图像是两条直线。

所以正确答案是 C。

Expanding, $x^2 + 2xy + y^2 = x^2 + y^2,$ which reduces to $2xy = 0,$ i.e. $xy = 0.$

This holds exactly when $x = 0$ or $y = 0,$ the two coordinate axes, so the graph is two lines.

Thus, the correct answer is C.