Let the equation of line passes through point
$\left(x_1, y_1\right)$ having slope $m$ is
$y-y_1-m\left(x-x_1\right)=0$
Now, algebraic sum of the lengths of the perpendiculars drawn to the line (i) from $(2,0)$, $(0,2)$ and $(1,1)$ is
$\begin{aligned} & \frac{-y_1-2 m+m x_1}{\sqrt{1+m^2}}+\frac{2-y_1+m x_1}{\sqrt{1+m^2}} \\ & +\frac{1-y_1-m+m x_1}{\sqrt{1+m^2}}=0 \text { (given) }\end{aligned}$
$\begin{aligned} \Rightarrow & & m\left(-2+x_1+x_1-1+x_1\right) \\ & & +\left(-y_1+2-y_1+1-y_1\right)=0 \\ \Rightarrow & & m\left(3 x_1-3\right)+\left(3-3 y_1\right)=0 \\ \Rightarrow & & m\left(x_1-1\right)+\left(1-y_1\right)=0\end{aligned}$
On comparing the coefficient of different terms, We get $x_1-1=0$ and $y_1-1=0$
So, $\left(x_1, y_1\right)=(1,1)$
Hence, option (a) is correct.