jmc

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 $y=-5\cdot 2+18=8$. Thus $(a,b)=(2,8)$, and $a+b=\boxed{10}$.