11

11. [Maximum mark: 19]
    The plane $\Pi_1$ has equation $2x + 6y - 2z = 5$.

(a) Verify that the point $A\!\left(2, \frac{1}{2}, 1\right)$ lies on the plane $\Pi_1$.

The plane $\Pi_2$ is given by $(k^2 - 6)x + (2k + 3)y + pz = q$, where $p, q, k \in \mathbb{R}$ and $p \neq 0$.

(b) In the case where $p = -6$, $\Pi_2$ is perpendicular to $\Pi_1$ and $A$ lies on $\Pi_2$. Find the value of $k$ and the value of $q$.

For parts (c), (d) and (e) it is now given that $\Pi_2$ is parallel to $\Pi_1$ with $k = 3$.

(c) Determine the value of $p$.

It is also given that $q = -\frac{51}{2}$. The line through $A$ that is perpendicular to $\Pi_1$ meets $\Pi_2$ at the point $B$.

(d) (i) Find the coordinates of $B$.

(ii) Hence, show that the perpendicular distance between $\Pi_1$ and $\Pi_2$ is $\sqrt{11}$.

(e) Find the equation of a third parallel plane $\Pi_3$ which is also a perpendicular distance of $\sqrt{11}$ from $\Pi_1$.