A piece of cheese is located at $(12,10)$ in a [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane). A mouse is at $(4,-2)$ and is running up the [line](https://artofproblemsolving.com/wiki/index.php/Line) $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is  $a+b$?
$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 14\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 22$
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 $a+b = \boxed{10}$.