2025 CJC P1 Q8

The plane \(p\) passes through the points \(A\), \(B\) and \(C\) with coordinates \((1, 0, 2)\), \((2, -1, 3)\) and \((-4, -1, 0)\) respectively.

1. Show that a cartesian equation of \(p\) is \(x - y - 2z = -3\).[3]

The line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix}, \lambda \in \mathbb{R}\).

2. Find the acute angle between \(l\) and \(p\).[2]

It is given that a variable point \(R\) lies on \(p\) and is at a distance of \(\sqrt{22}\) from the point \(Q\) with coordinates \((3, 4, -2)\).

3. Find the foot of perpendicular from \(Q\) to \(p\).[4]
4. Hence describe geometrically the path traced by \(R\).[2]