Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

General point on line $\mathrm{L}:\frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$
is $(\lambda +1,-\lambda -1,2\lambda +2)$

DR's of PM are $(\lambda ,-\lambda -3,2\lambda )$
$PM\perp L$
$\begin{array}{rl}& \Rightarrow \lambda +(-1)(-\lambda -3)+2(2\lambda )=0\\ & \Rightarrow 6\lambda +3=0\end{array}$

$P\left(\frac{1}{2},\frac{-1}{2},1\right)$

Let another line $L^{\prime }:\frac{x+1}{1}=\frac{y-1}{-1}=\frac{z+2}{1}$
General point on line $L^{\prime }$ is $(\mu -1,-\mu +1,\mu -2)$
Point of intersection of line $L$ and $L^{\prime }$ is
$\begin{array}{ll}\lambda +1=\mu -1& 2\lambda +2=\mu -2\\ \Rightarrow \mu -\lambda =2\dots (1)& \Rightarrow 2\lambda =\mu -4\end{array}$

$\begin{array}{rl}& Q(-1,1,-2)\\ & 2(PQ{)}^2=2\left({\left(\frac{1}{2}+1\right)}^2+{\left(\frac{-1}{2}-1\right)}^2+(1+2{)}^2\right)\\ & =2\left(\frac{9}{4}+\frac{9}{4}+9\right)\\ & =27\end{array}$