2007 AMC 12A 第 13 题

Question

一块奶酪位于坐标平面上的

(12,10)

。一只老鼠在

(4,-2)

，并沿直线

y=-5x+18

向上跑。在点

(a,b)

处，老鼠开始离奶酪越来越远，而不是越来越近。求

a+b

？

A piece of cheese is located at

(12,10)

in a coordinate plane. A mouse is at

(4,-2)

and is running up the line

y=-5x+18.

At the point

(a,b)

the mouse starts getting farther from the cheese rather than closer to it. What is

a+b?

6

10

14

18

22

答案：B

知识点：坐标几何斜率

难度评级：1410

小提示：

直线上距离奶酪最近的点是从奶酪向该直线作垂线的垂足。

The closest point on the line is the foot of the perpendicular from the cheese

大提示：

过

(12,10)

的垂线斜率为

\tfrac15

；求它与

y=-5x+18

的交点。

The perpendicular through

(12,10)

has slope

\tfrac15

; intersect it with

y=-5x+18

解答：

老鼠在从

(12,10)

到该直线的垂足处最接近奶酪。

这条垂线斜率为

\tfrac15

，所以方程为

y=10+\tfrac15(x-12)=\tfrac15 x+\tfrac{38}{5}

。

令

\tfrac15 x+\tfrac{38}{5}=-5x+18

，得

x=2

且

y=8

。因此

(a,b)=(2,8)

，

a+b=10

。

因此，正确答案是 B。

The mouse is closest to the cheese at the foot of the perpendicular from

(12,10)

to the line.

This perpendicular has slope

\tfrac15,

so its equation is

y=10+\tfrac15(x-12)=\tfrac15 x+\tfrac{38}{5}.

Setting

\tfrac15 x+\tfrac{38}{5}=-5x+18

gives

x=2

and

y=8.

Thus

(a,b)=(2,8)

and

a+b=10.

Thus, the correct answer is B.

2007 AMC 12A 第 13 题

答案：B

难度评级：1410

小提示：

大提示：

解答：

其他年份的第 13 题