AMC10 2006 A

AMC10 2006 A · Q11

AMC10 2006 A · Q11. It mainly tests Linear equations, Transformations.

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

以下哪个描述了方程 $(x+y)^2 = x^2 + y^2$ 的图像？

(A) the empty set 空集

(B) one point 一点

(D) a circle 一个圆

(E) the entire plane 整个平面

Answer

Correct choice: (C)

正确答案：（C）

Solution

The equation $(x + y)^2 = x^2 + y^2$ is equivalent to $x^2 + 2xy + y^2 = x^2 + y^2$, which reduces to $2xy = 0$, so $xy = 0$. Thus the graph consists of the two lines that are the coordinate axes.

方程 $(x + y)^2 = x^2 + y^2$ 等价于 $x^2 + 2xy + y^2 = x^2 + y^2$，化简为 $2xy = 0$，所以 $xy = 0$。因此图像是由两条坐标轴组成的直线。

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