AMC12 2006 A
AMC12 2006 A · Q11
AMC12 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
一个点
(C)
two lines
两条直线
(D)
a circle
一个圆
(E)
the entire plane
整个平面
Answer
Correct choice: (C)
正确答案：（C）
Solution
\begin{align*}(x+y)^2&=x^2+y^2\\ x^2 + 2xy + y^2 &= x^2 + y^2\\ 2xy &= 0\end{align*}
Either $x = 0$ or $y = 0$. The union of them is 2 lines, and thus the answer is $\mathrm{(C)}$.
\begin{align*}(x+y)^2&=x^2+y^2\\ x^2 + 2xy + y^2 &= x^2 + y^2\\ 2xy &= 0\end{align*}
因此要么 $x = 0$，要么 $y = 0$。它们的并集是两条直线，所以答案是 $\mathrm{(C)}$。
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