AMC10 2013 B

AMC10 2013 B · Q14

AMC10 2013 B · Q14. It mainly tests Linear equations, Transformations.

Define $a \ast b = a^2b - ab^2$. Which of the following describes the set of points $(x, y)$ for which $x \ast y = y \ast x$?

定义 $a \ast b = a^2b - ab^2$。以下哪项描述了满足 $x \ast y = y \ast x$ 的点集 $(x, y)$？

(A) a finite set of points 有限点集

(B) one line 一条直线

(D) two intersecting lines 两条相交直线

(E) three lines 三条直线

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): The equation $x\clubsuit y = y\clubsuit x$ is equivalent to $x^2y-xy^2 = y^2x-yx^2$. This equation is equivalent to gives $2xy(x-y)=0$. This equation will hold exactly if $x=0$, $y=0$, or $x=y$. The solution set consists of three lines: the $x$-axis, the $y$-axis, and the line $x=y$.

答案（E）：方程 $x\clubsuit y = y\clubsuit x$ 等价于 $x^2y-xy^2 = y^2x-yx^2$。该方程等价于 $2xy(x-y)=0$。该方程成立当且仅当 $x=0$、$y=0$ 或 $x=y$。解集由三条直线组成：$x$ 轴、$y$ 轴以及直线 $x=y$。

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