2019 CJC P1 Q9

Line \({{l}_{1}}\) has equation \(\mathbf{r}=\left( \begin{matrix} -1 \\ 8 \\ 3 \end{matrix}\right)+\lambda \left( \begin{matrix} 2 \\ 1 \\5 \end{matrix} \right)\) where \(\lambda \) is a real parameter, and plane \(p\) has equation \(\mathrm{r}\cdot \left( \begin{matrix}a \\ b \\0 \end{matrix} \right)=17\).

It is given that \({{l}_{1}}\) lies completely on \(p\) and the point \(Q\) has coordinates \((-1,-2,3)\).

1. Show that \(a=-1\) and \(b=2\).[3]
2. Find the foot of perpendicular from point \(Q\) to \(p\). Hence find the shortest distance between point \(Q\) and \(p\) in exact form.[4]

Given that the shortest distance between \({{l}_{1}}\) and the foot of perpendicular of \(Q\) onto \(p\) is \(\frac{5}{3}\sqrt{6}\).

3. Using the result obtained in part (ii) or otherwise, find the shortest distance between \(Q\) and \({{l}_{1}}\).[2]

The line \({{l}_{2}}\) is parallel to \(p\), and has a direction perpendicular to \({{l}_{1}}\) and passes through point \(Q\).

4. Show that the Cartesian equation of the line \({{l}_{2}}\) is \(\frac{x+1}{2}=y+2=3-z\).[3]
5. Find the vector equation of line \({{l}_{3}}\), which is the reflection of \({{l}_{2}}\) about \(p\).[3]