The foot of the perpendicular from the point  $\left(1,2\right)$ on the line passing through the points $\left(5,4\right)$ and $\left(7,4\right)$ is $\left(\mathrm{a},\mathrm{b}\right)$ then  $\mathrm{a}+\mathrm{b}$

Equation of the line joining two points $\left(5,4\right)$ and $\left(7,4\right)$ is $y=4$ , because y coordinates are same and equal to 4.
Foot of the perpendicular from the point $\left(1,2\right)$ on the line $y=4$ is $\left(1,4\right)$ , which is   $\left(\mathrm{a},\mathrm{b}\right)$
Hence  $\mathrm{a}+\mathrm{b}=1+4=\boxed{5}$