N2016 P1 Q11

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

1. In the case where \(a=0\),
   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 \(12\).[5]
2. Find the value of \(a\) such that \(l\) and \(p\) do not meet in a unique point.[3]