2006 AMC 10A 第 11 题

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

(x + y)^2

并化简。

Expand

(x + y)^2

and simplify

方程化简为

xy = 0

。

The equation reduces to

xy = 0

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.

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