2022 YIJC P2 Q5

The plane \(p\) has equation \(\mathbf{r}=\left( \begin{matrix} 6 \\ 2 \\ -5 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 0 \\ 3 \\ -1 \\ \end{matrix} \right)+\mu \left( \begin{matrix} 2 \\ -2 \\ 1 \\ \end{matrix} \right)\), where \(\lambda \) and \(\mu \) are parameters.

The line \(l\) passes through the points \(A\) and \(B\) with position vectors \(4\mathbf{i}+2\mathbf{j}-\mathbf{k}\) and \(4\mathbf{j}+2\mathbf{k}\) respectively.

1. Find the coordinates of the point of intersection between \(p\) and \(l\).[5]
2. Find the cartesian equations of the planes such that the perpendicular distance from each plane to \(p\) is \(\sqrt{41}\).[3]

Another line \(m\) has equation \(\frac{1-x}{3}=y+2=\frac{z-3}{a}\).

3. Find the value of a such that \(p\) and \(m\) do not meet in a unique point.[3]