2019 JPJC P2 Q5

The plane \(p\) has equation \(\mathbf{r}=\left( \begin{matrix} 1 \\ 0 \\ -3 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 2 \\ 0 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 2 \\ 3 \\ -4 \\ \end{matrix} \right)\), and the line \(l\) has equation \(\mathbf{r}=\left( \begin{matrix} -10 \\ 0 \\ 4 \\ \end{matrix} \right)+t\left( \begin{matrix} 8 \\ -4 \\ 1 \\ \end{matrix} \right)\), where \(\lambda \), \(\mu \) and \(t\) are parameters.

1. Show that \(l\) is perpendicular to \(p\) and find the values of \(\lambda \), \(\mu \) and \(t\) which give the coordinates of the point at which \(l\) and \(p\) intersect.[5]
2. Find the cartesian equations of the planes such that the perpendicular distance from each plane to \(p\) is \(2\).[5]