2023 ASRJC P1 Q4

The equations of planes \({{p}_{1}}\) and \({{p}_{2}}\) are respectively given by

\(\mathbf{r}=s\left( \begin{matrix} \begin{matrix} 0 \\ 1 \\ 1 \\ \end{matrix} \\ \end{matrix} \right)+t\left( \begin{matrix} \begin{matrix} 3 \\ -1 \\ 2 \\ \end{matrix} \\ \end{matrix} \right)\), \(s\), \(t\in \mathbb{R}\) and \(\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} \alpha \\ 3 \\ \beta \\ \end{matrix} \\ \end{matrix} \right)=7\),

where \(\alpha \) and \(\beta \) are real constants.

1. Given that the line \({{l}_{1}}:\frac{x+1}{4}=\frac{2-y}{3}=z\) lies in \({{p}_{2}}\), find \(\alpha \) and \(\beta \).[3]

It is now given that \(\alpha =-3\) and \(\beta =2\).

2. Find the equation of the line of intersection of \({{p}_{1}}\) and \({{p}_{2}}\).[3]
3. The plane \({{p}_{3}}\) with equation \(y+3z+4=0\) is parallel to \({{l}_{1}}\). Find the distance between \({{l}_{1}}\) and \({{p}_{3}}\).[2]