AMC12 2007 A

AMC12 2007 A · Q13

AMC12 2007 A · Q13. It mainly tests Coordinate geometry, Distance / midpoint.

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 $a+b$?

坐标平面上，一块奶酪位于 $(12,10)$。一只老鼠在 $(4,-2)$，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) 6 6

(B) 10 10

(D) 18 18

(E) 22 22

Answer

Correct choice: (B)

正确答案：（B）

Solution

The point $(a,b)$ is the foot of the perpendicular from $(12,10)$ to the line $y=-5x+18$. The perpendicular has slope $\frac{1}{5}$, so its equation is $y=10+\frac{1}{5}(x-12)=\frac{1}{5}x+\frac{38}{5}$. The $x$-coordinate at the foot of the perpendicular satisfies the equation $\frac{1}{5}x+\frac{38}{5}=-5x+18$, so $x=2$ and $y=-5\cdot2+18=8$. Thus $(a,b) = (2,8)$, and $a+b = \boxed{10}$.

点 $(a,b)$ 是从 $(12,10)$ 向直线 $y=-5x+18$ 作垂线的垂足。垂线斜率为 $\frac{1}{5}$，所以其方程为 $y=10+\frac{1}{5}(x-12)=\frac{1}{5}x+\frac{38}{5}$。垂足的 $x$ 坐标满足 $\frac{1}{5}x+\frac{38}{5}=-5x+18$，所以 $x=2$ 且 $y=-5\cdot2+18=8$。因此 $(a,b) = (2,8)$，并且 $a+b = \boxed{10}$。

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