2016 AMC 12A 第 7 题

Question

下列哪一项描述了

x^2(x+y+1)=y^2(x+y+1)

的图像？

Which of these describes the graph of

x^2(x+y+1)=y^2(x+y+1)?

两条平行直线

two parallel lines

两条相交直线

two intersecting lines

三条直线都经过同一个公共点

three lines that all pass through a common point

三条直线不全经过同一个公共点

three lines that do not all pass through a common point

一条直线和一条抛物线

a line and a parabola

答案：D

知识点：因式分解平方差

难度评级：1410

小提示：

把所有项移到一边，并提出

x+y+1

。

Move everything to one side and factor out

x+y+1

大提示：

(x-y)(x+y)(x+y+1)=0

是三条直线的并。

(x-y)(x+y)(x+y+1)=0

is the union of three lines

解答：

将所有项移到一边，得到

(x^2-y^2)(x+y+1)=0\text{，}

再分解为

(x-y)(x+y)(x+y+1)=0\text{。}

因此图像是直线

x=y

、

x=-y

和

x+y+1=0

的并集。

前两条直线交于原点，但第三条直线

x+y=-1

与

x+y=0

平行，不经过原点。因此，图像由三条不全经过同一个公共点的直线组成。

所以正确答案是 D。

Moving all terms to one side gives

(x^2-y^2)(x+y+1)=0,

which factors as

(x-y)(x+y)(x+y+1)=0.

The graph is therefore the union of the lines

x=y,

x=-y,

and

x+y+1=0.

The first two lines intersect at the origin, but the third line

x+y=-1

is parallel to

x+y=0

and does not pass through the origin. So the graph consists of three lines that do not all pass through a common point.

Thus, the correct answer is D.

2016 AMC 12A 第 7 题

答案：D

难度评级：1410

小提示：

大提示：

解答：

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