From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars $PQ$ and $\mathrm{PR}$ are drawn respectively on the lines $y=x,z=1$  and $y=-x,z\mathit{=}\mathit{-}\mathit{1}$

If $\mathrm{P}$ is such that $\mathrm{\angle }QPR$ is a right angle, then the possible value(s) of $\lambda$ is: (are)

Line ${\mathrm{L}}_1$ given by $y=x,z=1$ can be expressed $L_1:\frac{x}{1}=\frac{y}{1}=\frac{z-1}{0};\frac{x}{1}=\frac{y}{1}=\frac{z-1}{0}=\alpha$ (say)

$\Rightarrow x=\alpha ,y=\alpha ,z=1$

Let the coordinates of $\mathrm{Q}$ on ${\mathrm{L}}_1$ be $( \mathrm{a},\mathrm{a}, \mathrm{l})$

line ${\mathrm{L}}_2$ given by $y=-x,z=-1$ can be expressed as 

$L_2:\frac{x}{1}=\frac{y}{-1}=\frac{z+1}{0}\frac{x}{1}=\frac{y}{-1}=\frac{z+1}{0}=\beta$ (say)

$\Rightarrow x=\beta ,y=-\beta ,z=-1$

Let the coordinates of $\mathrm{R}$ on ${\mathrm{L}}_2$ be $(\beta ,-\beta ,-1)$

Direction ratios of $\mathrm{PQ}$ are $\lambda -\alpha ,\lambda -\alpha ,\lambda -1$.

Now. $PQ\perp L_1$

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1(\lambda -\alpha )+1\cdot (\lambda -\alpha )+0\cdot (\lambda -1)=0\Rightarrow \lambda =\alpha \\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}Q(\lambda ,\lambda ,1)\end{array}$

Direction ratio of $\mathrm{PR}$ are $\lambda -\beta ,\lambda +\beta ,\lambda +1$.

Now,$PR\perp L_2$

$\begin{array}{l}\therefore 1(\lambda -\beta )+(-1)(\lambda +\beta )+0(\lambda +1)=0\\ \lambda -\beta -\lambda -\beta =0\Rightarrow \beta =0\end{array}$

$\therefore R(0,0,-1)$

Now, as $\mathrm{\angle }QPR={90}^{\circ }$

(as $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$, if two lines with DR's $a_1,b_1,c_1;a_2,b_2,c_2$ are perpendicular) 

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(\lambda -\lambda )(\lambda -0)+(\lambda -\lambda )(\lambda -0)+(\lambda -1)(\lambda +1)=0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(\lambda -1)(\lambda +1)=0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\lambda =1\end{array}$

or $\lambda =-1$

$\lambda =1$ rejected as $\mathrm{P}$ and $\mathrm{Q}$ are different points. 

$\Rightarrow \lambda =-1$