Since, distance $\mathrm{P}\mathrm{Q}$ is measured parallel to the line
$20\mathrm{x}=5\mathrm{y}=4\mathrm{z}$ i.e. $\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{4}=\frac{\mathrm{z}}{5}$ .
$\Rightarrow$ Direction ratios of line $\mathrm{P}\mathrm{Q}$ are $1,\mathrm{}4,\mathrm{}5$

Equation of $\mathrm{P}\mathrm{Q}$
$\Rightarrow \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}+2}{4}=\frac{\mathrm{z}-3}{5}=\mathrm{\lambda },$ (say)
$\Rightarrow$ A point on the line is $\left(\mathrm{\lambda }+\mathrm{1,4}\mathrm{\lambda }-\mathrm{2,5}\mathrm{\lambda }+3\right)\equiv \mathrm{M}$ , (say)
Point $\mathrm{}\mathrm{M}\mathrm{}$ lies on the plane $2\mathrm{x}+3\mathrm{y}-4\mathrm{z}+22=0$
$\Rightarrow 2\left(\mathrm{\lambda }+1\right)+3\left(4\mathrm{\lambda }-2\right)-4\left(5\mathrm{\lambda }+3\right)+22=0$
$\iff -6\mathrm{\lambda }+6=0\Rightarrow \mathrm{\lambda }=1$
$\Rightarrow \mathrm{M}=\left(\mathrm{2,2},8\right)$
$\mathrm{M}$ is the midpoint of $\mathrm{P}\mathrm{Q}$
$\Rightarrow \left|\text{PQ}\right|=2\left|\text{PM}\right|$
$\Rightarrow \left|\text{PQ}\right|=2\sqrt{{\left(2-1\right)}^2+{\left(2+2\right)}^2+{\left(8-3\right)}^2}$
$=2\sqrt{1+16+25}$
$=2\sqrt{42}$
$\Rightarrow |\text{PQ}{|}^2=4\times 42=168$