Reflection of the midpoint of the line segment joining the points $\left(1,2\right)$ and  $\left(3,4\right)$ with respect to the line $y=x$ is $\left(a,b\right)$ . The value of $\mathrm{a}+\mathrm{b}$ is

Suppose that the midpoint of the line segment joining the points $\left(1,2\right)$ and $\left(3,4\right)$  is  P
Hence, 

$\begin{array}{c}\mathrm{P}=\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2}{2}\right)\\ =\left(\frac{1+3}{2},\frac{2+4}{2}\right)\\ =\left(2,3\right)\end{array}$
 The image of the point $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$  in the line y=x is  $\left({\mathrm{y}}_1,{\mathrm{x}}_1\right)$
Therefore, the image of the point  $P\left(2,3\right)$ in the line $y=x$ is  $\left(\mathrm{a},\mathrm{b}\right)=\left(3,2\right)$
It implies that   $\mathrm{a}+\mathrm{b}=3+2=\boxed{5}$