2023 EJC P2 Q5

Planes \(p\) and \(q\) are perpendicular. Plane \(p\) has equation \(2x-z=10\). Plane \(q\) contains the line \(l\) with equation
\(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 5 \\ -3 \\ 4 \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} 2 \\ -2 \\ 1 \\ \end{matrix} \\ \end{matrix} \right)\), where \(\lambda \) is a parameter.

1. Show that \(q\) has equation \(x+2y+2z=7\).[2]
2. Find a vector equation of the line \(m\) where \(p\) and \(q\) meet.[2]

The point \(A\) with coordinates \(\left( 2,\,1,\,-6 \right)\) lies on \(p\) and the shortest distance from \(A\) to \(q\) is \(k\).

3. Find the position vector of the foot of perpendicular, \(F\) from \(A\) to \(q\). Hence find the value of \(k\).[4]
4. Find vector equations of the lines in \(p\) such that the shortest distance from each line to \(q\) is \(k\).[3]