DR's of Line $L \equiv-1: 1: 2$
DR's of $A B \equiv \alpha-2: \beta-2: \gamma-2$
$
\begin{aligned}
& \mathrm{AB} \perp_{\mathrm{ar}} \mathrm{~L} \Rightarrow 2-\alpha+\beta-2+2 \gamma-4=0 \\
& 2 \gamma+\beta-\alpha=4
\end{aligned}
$
Let C is mid-point of AB
$
\mathrm{C}\left(\frac{\alpha+2}{2}, \frac{\beta+2}{2}, \frac{\gamma+2}{2}\right)
$
DR's of $\mathrm{PC}=\frac{\alpha}{2}: \frac{\beta-2}{2}: \frac{\gamma}{2}$
line $\mathrm{L} \| \mathrm{PC} \Rightarrow \frac{-\alpha}{2}=\frac{\beta-2}{2}=\frac{\gamma}{4}=\mathrm{K}$ (let)
$
\begin{aligned}
& \alpha=-2 \mathrm{~K} \\
& \beta=2 \mathrm{~K}+2 \\
& \gamma=4 \mathrm{~K}
\end{aligned}
$
use in (1) ⇒ K = $\frac{1}{6}$
value of $\alpha+\beta+6 \gamma=24 K+2=6$
