2024 RI Promo Q9

The planes \(p\) and \(q\) have equations

\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]

where \(a\) and \(b\) are constants and \(\lambda\) and \(\mu\) are parameters.

The line \(l\) passes through the point \((5, 4, 0)\) and is parallel to the vector \(-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). The planes \(p\) and \(q\) meet in the line \(l\).

1. Show that \(a = 1\) and \(b = \frac{1}{2}\).[2]
2. Find the exact acute angle between the planes \(p\) and \(q\).[3]
3. Find the distance from the point \(A(2, 0, 3)\) to the plane \(q\). Hence deduce the shortest distance from \(A\) to \(l\).[4]

The plane \(q\) is reflected about the plane \(p\) to obtain the plane \(q'\).

4. Find a cartesian equation of the plane \(q'\).[3]