$\text{ If the line }4x-5y+41=0\text{ is the perpendicular bisector of the line segment joining the }$

$\text{ points }A(1,1)\text{ and }B\text{. The image of }B\text{ in }x\text{-axis is }(a,b)\text{ then the numerical value of a+b is}$

$\text{ The point }B\text{ is the image of }A(1,1)\text{ with respect to the line }4x-5y+41=0$

$\text{ If }(h,k)\text{ is the image of the point }\left(x_1,y_1\right)\text{ with respect to the line }ax+by+c=0\text{ then }$

$\frac{h-x_1}{a}=\frac{k-y_1}{b}=-\frac{2\left(a{x}_1+b{y}_1+c\right)}{a^2+b^2}$

Hence, 

$\frac{h-1}{4}=\frac{k-1}{-5}=-\frac{2\left(4-5+41\right)}{4^2+{\left(-5\right)}^2}$

Simplify 

  $\frac{h-1}{4}=\frac{k-1}{-5}=-\frac{80}{41}$          

Hence,

    $\begin{array}{c}\frac{h-1}{4}=-\frac{80}{41}\\ h=\frac{-320}{41}+1\\ =\frac{-320+41}{41}\\ =-\frac{279}{41}\end{array}$       $\begin{array}{c}\frac{k-1}{-5}=-\frac{80}{41}\\ k=\frac{400}{41}+1\\ =\frac{441}{41}\end{array}$

$\text{ Hence },(h,k)=\left(-\frac{279}{41},\frac{441}{41}\right)$

$\text{ The image of }B(h,k)\text{ with respect to }x-\text{ axis is }(h,-k)$ = (a,b)

$\begin{array}{c}a+b=h-k\\ =\frac{-441-279}{41}\\ =-\frac{720}{41}\\ =-17.56\end{array}$

$\text{ Therefore, }a+b=-17.56$