$$\begin{aligned} & A(2,2,2) \\ & P Q: \frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{-2} \end{aligned}$$

General point,

$$(k+1,-k+2,-2 k+1)$$

$$\begin{aligned} & \overrightarrow{O A}=(k+1-2) \hat{i}+(-k+2-2) \hat{j}+(-2 k+1-2) \hat{k} \\ & \overrightarrow{O A}=(k-1) \hat{i}-k \hat{j}+(-2 k-1) \hat{k} \\ & \overrightarrow{P Q}=1 \hat{i}-\hat{j}-2 \hat{k} \\ & \overrightarrow{O A} \cdot \overrightarrow{P Q}=0 \\ & (k-1)+k+2(2 k+1)=0 \\ & k-1+k+4 k+2=0 \\ & 6 k+1=0 \\ & k=\frac{-1}{6} \\ & 0\left(\frac{-1}{6}+1, \frac{+1}{6}+2,-2\left(\frac{-1}{6}\right)+1\right) \\ & 0\left(\frac{5}{6}, \frac{13}{6}, \frac{-8}{6}\right) \\ & 0\left(\frac{5}{6}, \frac{13}{6}, \frac{8}{6}\right)=\left(\frac{\alpha+2}{2}, \frac{\beta+2}{2}, \frac{\gamma+2}{2}\right) \end{aligned}$$

$$\begin{array}{l|l} \alpha+2=\frac{10}{6} & \beta+2=\frac{26}{6} \\ \alpha=\frac{10}{6}-2 & \beta=\frac{26-12}{6} \\ \alpha=\frac{-2}{6} & \beta=\frac{14}{6} \\ \alpha =-\frac{1}{3} & \beta=\frac{7}{3} \end{array}$$

$$\begin{aligned} & \gamma+2=\frac{16}{6} \\ & \gamma=\frac{16-12}{6} \\ & \gamma=\frac{4}{6} \\ & \Rightarrow \alpha+\beta+6 \gamma \\ & \Rightarrow \frac{-1}{3}+\frac{7}{3}+6 \times \frac{4}{6} \\ & \Rightarrow 2+4=6 \end{aligned}$$