1961 AMC 12 第 3 题

Question

若

2y+x+3=0

与

3y+ax+2=0

的图像垂直相交，则

a

的值为：

If the graphs of

2y+x+3=0

and

3y+ax+2=0

are to meet at right angles, the value of

a

is:

\pm\dfrac23

-\dfrac23

-\dfrac32

6

-6

将两个方程都化为斜截式

Put both equations into slope-intercept form

两条互相垂直的非竖直直线斜率之积为

-1

The product of the slopes of perpendicular nonvertical lines is

-1

两条直线的斜率分别为

-\frac{1}{2}

和

-\frac{a}{3}

。垂直条件要求

\left(-\frac12\right)\left(-\frac a3\right)=-1\text{，}

因而

a=-6

。

因此，正确答案是 E。

The two slopes are

-\frac{1}{2}

and

-\frac{a}{3}.

Perpendicularity requires

\left(-\frac12\right)\left(-\frac a3\right)=-1,

so

a=-6.

Therefore, the correct answer is E.

其他年份的第 3 题