2021 DHS P1 Q9

Points \(A\) and \(B\) have position vectors \(\left( \begin{matrix} 1 \\ q \\ 2 \\ \end{matrix} \right)\) and \(\left( \begin{matrix} -1 \\ 0 \\ 1 \\ \end{matrix} \right)\) respectively, where \(q\) is a positive constant.

The equation of the line \(l\) is \(\frac{4x-15}{-2}=y\), \(z=\frac{5}{2}\).

1. Given that the length of projection of vector \(\overrightarrow{AB}\) onto line \(l\) is \(\frac{8}{\sqrt{5}}\), show that \(q=5\).[3]
2. Find the coordinates of the point of \(A'\), where \(A'\) is the image of point \(A\) reflected in the line \(l\).[4]
3. The planes \({{p}_{1}}\) and \({{p}_{2}}\) have equations \(\mathbf{r}\cdot \left( \begin{matrix} 2 \\ 1 \\ k \\ \end{matrix} \right)=1\) and \(\mathbf{r}=\mu \left( \begin{matrix} 1 \\ s \\ 1 \\ \end{matrix} \right)+\gamma \left( \begin{matrix} 2 \\ -1 \\ 1 \\ \end{matrix} \right)\) respectively, for constants \(k\), \(s\) and parameters \(\mu \), \(\gamma \). Given that the two planes \({{p}_{1}}\) and \({{p}_{2}}\) intersect at \(l\), determine the values of \(k\) and \(s\).[3]
4. The plane \({{p}_{3}}\) has equation \(2x+ty+z=u\). Given that the planes \({{p}_{1}}\),\({{p}_{2}}\) and \({{p}_{3}}\) have no point in common, what can be said about the values of \(t\) and \(u\)?[2]