Let $2a+2b+c=0$, then the equation of the straight line $ax+by+c=0$ which is farthest from the point $\left(\mathrm{1,1}\right)$ is

All the lines $ax+by+c=0$ pass through the point $\left(\mathrm{2,2}\right).$
The line passing through $\left(\mathrm{2,2}\right)$ & farthest from $\left(\mathrm{1,1}\right)$ is perpendicular to the line joining $\left(\mathrm{1,1}\right)$ & $\left(\mathrm{2,2}\right)$
Now, the slope of the line joining $\left(\mathrm{1,1}\right)$ & $\left(\mathrm{2,2}\right)$ is $\frac{2-1}{2-1}=1$
$\Rightarrow$ The slope of the required line is $-1$
$\Rightarrow$ Required line is $\frac{y-2}{x-2}=-1$
$\Rightarrow y+x=4$