Question

定义 $a\clubsuit b=a^2b-ab^2$ 。下列哪一项描述了满足 $x\clubsuit y=y\clubsuit x$ 的点 $(x, y)$ 的集合？

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

有限个点

A finite set of points

一条直线

One line

两条平行直线

Two parallel lines

两条相交直线

Two intersecting lines

三条直线

Three lines

展开自定义运算两边。

Expand both custom-operation expressions

把得到的方程因式分解成直线因子。

Factor the resulting equation into line factors

条件 $x\clubsuit y=y\clubsuit x$ 展开为 $x^2y-xy^2=y^2x-yx^2$ 。

合并同类项并因式分解得 $2x^2y-2xy^2=2xy(x-y)=0$ 。

因此 $x=0$ 、 $y=0$ 或 $x=y$ ，这是三条直线。

所以正确答案是 E。

The condition is $x\clubsuit y=y\clubsuit x$ , so $x^2y-xy^2=y^2x-yx^2$ .

Combining like terms gives $2x^2y-2xy^2=2xy(x-y)=0$ .

Thus $x=0$ , $y=0$ , or $x=y$ . These are three lines.

Thus, the correct answer is E .