$P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ 

$\begin{aligned} & P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15 \\\\ & \text { then } \cos \theta=\frac{3}{\sqrt{6} \cdot \sqrt{6}}=\frac{1}{2} \\\\ & \begin{aligned} & \therefore \theta=\frac{\pi}{3}, \text { Now } \alpha=\frac{\pi}{2}-\theta \\\\ & \therefore \tan ^{2} \theta \cdot \cot ^{2} \alpha=\tan ^{4} \theta \\\\ &=(\sqrt{3})^{4}=9 \end{aligned} \end{aligned}$