2007 AMC 12A Problem 13

Question

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

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

The perpendicular through

(12,10)

has slope

\tfrac15

; intersect it with

y=-5x+18

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.

Problem 13 in other years