2019 TMJC P2 Q3

The plane \({{p}_{1}}\) contains the point \(A\) with coordinates \(\left( 1,\,\,2,\,\,8 \right)\) and the line \(l\) with equation \(\mathbf{r}=\left( \begin{matrix} 1 \\ -4 \\ 2 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 1 \\ -2 \\ \end{matrix} \right)\), where \(\lambda \) is a real parameter.

1. Show that a cartesian equation of plane \({{p}_{1}}\) is \(3x-y+z=9\).[2]

The foot of perpendicular from point \(A\) to line \(l\) is denoted as point \(F\).

2. Find the coordinates of point \(F\).[3]
3. Point \(B\) has coordinates \(\left( 1,-4,2 \right)\). Find the exact area of triangle \(ABF\).[2]
4. Point \(C\) has coordinates \(\left( -1,6,6 \right)\). By finding the shortest distance from point \(C\) to \({{p}_{1}}\), find the exact volume of tetrahedron \(ABFC\).[4]

[Volume of tetrahedron\(=\frac{1}{3}\times \mathrm{base}\,\,\mathrm{area}\times \mathrm{perpendicular height}\)]

5. Point \(D\) lies on the line segment \(AC\) such that \(AD:DC=1:3\). Another plane \({{p}_{2}}\) is parallel to \({{p}_{1}}\) and contains point \(D\). Find a cartesian equation of \({{p}_{2}}\).[2]