Problem:
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?
Answer Choices:
A. 6
B. 10
C. 14
D. 18
E. 22
Solution:
The point (a,b) is the foot of the perpendicular from (12,10) to the line y=−5x+18. The perpendicular has slope 51, so its equation is
y=10+51(x−12)=51x+538
The x-coordinate at the foot of the perpendicular satisfies the equation
51x+538=−5x+18
so x=2 and y=−5⋅2+18=8. Thus (a,b)=(2,8), and a+b=10.
OR
If the mouse is at (x,y)=(x,18−5x), then the square of the distance from the mouse to the cheese is
(x−12)2+(8−5x)2=26(x2−4x+8)=26((x−2)2+4).
The value of this expression is smallest when x=2, so the mouse is closest to the cheese at the point (2,8), and a+b=2+8=10.
The problems on this page are the property of the MAA's American Mathematics Competitions