2026 EJC P1 Q10

The plane \(p\) has equation \(\mathbf r=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\0\end{pmatrix}+\mu\begin{pmatrix}-1\\0\\1\end{pmatrix}\), where \(\lambda,\mu\) are parameters.

1. Use a vector product to show that \(2\mathbf i-\mathbf j+2\mathbf k\) is normal to \(p\). Hence find the cartesian equation of \(p\).[3]

The line \(l\) has equation \(\mathbf r=\begin{pmatrix}t\\1\\-2\end{pmatrix}+s\begin{pmatrix}-t\\t^2\\2t+4\end{pmatrix}\), where \(s\) is a parameter and \(t\) is an unknown constant.

2. Show that, for all values of \(t\), \(l\) is not perpendicular to \(p\).[3]
3. Given that \(l\) does not intersect \(p\), show that \(t=-2\).[4]
4. For \(t=-2\), find the shortest distance between \(l\) and \(p\).[3]