The distance from the origin to the image of ( 1, 1) with respect to the line x+y+5=0

Let (h, k) be the image of (1, 1) w.r.to x+y+5=0

$\begin{array}{l}\therefore \frac{h-x_1}{a}=\frac{k-y_1}{b}=\frac{-2\left(a{x}_1+b{y}_1+c\right)}{a^2+b^2}\\ \Rightarrow \frac{h-1}{1}=\frac{k-1}{1}=\frac{-2\left(1+1+5\right)}{1+1}\\ \Rightarrow \frac{h-1}{1}=\frac{k-1}{1}=-7\end{array}$

$\begin{array}{l}\therefore h-1=-7 k-1=-7\\ \Rightarrow h=-7+1k=-7+1\\ \Rightarrow h=-6 k=-6\end{array}$

$\begin{array}{l}\therefore P\left(h,k\right)=\left(-6,-6\right)andO\left(0,0\right)\\ OP=\sqrt{{\left(0+6\right)}^2+{\left(0+6\right)}^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}\end{array}$