2023 NJC P1 Q8

A tetrahedron \(OABC\) has four triangular faces with a vertex \(O\) at the origin. The coordinates of \(A\), \(B\) and \(C\) are \(\left( 9,0,0 \right)\), \(\left( 0,2,0 \right)\) and \(\left( 0,0,3 \right)\) respectively (see diagram).

1. Find a cartesian equation of face \(ABC\).[3]
2. Find the acute angle between face \(ABC\) and the \(z\)-axis.[2]
3. Find the coordinates of the foot of perpendicular from \(O\) to face \(ABC\).[2]
4. The line \(s\) passes through \(O\) and the midpoint of \(AC\). Show that a vector equation of the reflection of \(s\) in face \(ABC\), is \(\mathbf{r}=\left( \begin{matrix} \frac{9}{2} \\ 0 \\ \frac{3}{2} \\ \end{matrix} \right)+\mu \left( \begin{matrix} a \\ -216 \\ -23 \\ \end{matrix} \right)\), where \(a\) is a constant to be determined.[3]