Mathematics Question from JEE Main 2023 · 31 January, Shift 1

Let $\theta$ be the angle between the planes

$$P_1: \vec{r} \cdot ( \hat{i} + \hat{j} + 2 \hat{k}) = 9 \quad \text{and} \quad P_2: \vec{r} \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 15.$$

Let $L$ be the line that meets $P_2$ at the point $(4, -2, 5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then

$$(\tan^2 \theta)(\cot^2 \alpha)$$

is equal to: