2021 TMJC P1 Q9

The line \(l\) passes through the point \(A\) with coordinates \(\left( 1,-2,3 \right)\) and is parallel to the vector \(\left( \begin{matrix} 4 \\ 0 \\ -1 \\ \end{matrix} \right)\).

The plane \({{\pi }_{1}}\) contains the point \(B\) with coordinates \(\left( 2,-1,3 \right)\) and the line \(l\).

1. Show that the cartesian equation of \({{\pi }_{1}}\) is \(x-y+4z=15\).[2]

Let \(F\) be the point on \(l\) which is closest to \(B\).

2. Find the coordinates of \(F\).[3]

Point \(C\) has coordinates \(\left( 8,1,-3 \right)\).

3. Find the exact shortest distance from \(C\) to \({{\pi }_{1}}\).[2]
4. Hence, find the exact volume of the tetrahedron \(CABF\).[3]

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

5. \(D\) is a general point on plane \({{\pi }_{2}}\) such that \(DABF\) has the same volume as tetrahedron \(CABF\). Given that \({{\pi }_{2}}\) does not contain \(C\), find an equation of \({{\pi }_{2}}\).[2]