Let $Q=\left(\lambda ,1,-\lambda -1\right)$

Direction ratio's of $PQ\equiv \left(\lambda -\beta ,1-0, -\lambda -1-\beta \right)$

$\therefore PQ$ is perpendicular to given line, hence

$1\left(\lambda -\beta \right)+0\left(1\right)-1\left(-\lambda -1-\beta \right)=0$

$\Rightarrow \left(\lambda -\beta \right)+\lambda +1+\beta =0$

$\Rightarrow \lambda =-\frac{1}{2}$

$\therefore PQ= \sqrt{{\left(\lambda -\beta \right)}^2+1^2+{\left(-\lambda -1-\beta \right)}^2}=\sqrt{\frac{3}{2}}$

$\Rightarrow {\left(-\frac{1}{2}-\beta \right)}^2+1^2+{\left(\frac{1}{2}-1-\beta \right)}^2=\frac{3}{2}$

$\Rightarrow {\beta }^2+\frac{1}{4}+\beta +1+{\beta }^2+\frac{1}{4}+\beta =\frac{3}{2}$

$\Rightarrow 2{\beta }^2+2\beta =0\Rightarrow \beta =0$ or $-1$