2021 NYJC P2 Q1

The plane \({{\prod }_{1}}\) passes through \(\left( 3,-1,2 \right)\) and is perpendicular to the line \(\mathbf{r}=2\mathbf{i}+4\mathbf{j}-3\mathbf{k}+\lambda \left( 5\mathbf{i}+2\mathbf{j}-\mathbf{k} \right)\). The plane \({{\prod }_{2}}\) contains the points \(\left( 2,3,2 \right)\), \(\left( 4,1,-1 \right)\) and \(\left( 0,-1,2 \right)\).

1. Show that the acute angle, \(\theta \), between the planes \({{\prod }_{1}}\) and \({{\prod }_{2}}\) is such that \(\cos \theta =\frac{\sqrt{30}}{15}\).[3]
2. Show that the line of intersection, \(L\), of the planes \({{\prod }_{1}}\) and \({{\prod }_{2}}\) has vector equation \(\mathbf{r}=\left( \begin{matrix}0 \\9 \\7 \end{matrix} \right)+\mu \left(\begin{matrix}-1 \\4 \\3 \end{matrix} \right)\), \(\mu \in \mathbb{R}\).[3]

The plane \({{\prod }_{3}}\) has the equation \(4\left( k-2 \right)x+\left( k+1 \right)y-4{{k}^{2}}z=8\), where \(k\) is a constant.

3. The three planes \({{\prod }_{1}}\),\({{\prod }_{2}}\) and \({{\prod }_{3}}\)have no points in common. By considering the relationship between the line \(L\) and the plane \({{\prod }_{3}}\), find the possible values of \(k\).[2]
4. For the positive value of \(k\) found in (iii), find the distance between \(L\) and \({{\prod }_{3}}\).[2]