2020 TMJC Promo Q8

The plane \(\pi \) contains the point \(A\) with coordinates \(\left( -2,1,4 \right)\) and the line with equation \(\mathbf{r}=\left( \begin{matrix} 0 \\ 3 \\ 2 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} -2 \\ 0 \\ 1 \\ \end{matrix} \right)\), where \(\lambda \) is a parameter.

1. Show that the cartesian equation of \(\pi \) is \(x+y+2z=7\).[2]

Let \(L\) be the set of lines such that the equation of any line \({{l}_{a}}\) in \(L\) is given by \(\mathbf{r}=\left( \begin{matrix} -2a-2 \\ 1 \\ 4a+2 \\ \end{matrix} \right)+\mu \left( \begin{matrix} -4-a \\ 1 \\ 2a+3 \\ \end{matrix} \right)\), where \(\mu \) is a parameter and \(a\) is a real constant.

2. Verify that the point \(B\) with coordinates \(\left( 6,-1,-4 \right)\) lies on any line \({{l}_{a}}\) in \(L\).[2]

The equation of a particular line \({{l}_{c}}\) in \(L\) is given by \(\mathbf{r}=\left( \begin{matrix} -2c-2 \\ 1 \\ 4c+2 \\ \end{matrix} \right)+\mu \left( \begin{matrix} -4-c \\ 1 \\ 2c+3 \\ \end{matrix} \right)\), where \(\mu \) is a parameter and \(c\) is a real constant. It is given that \({{l}_{c}}\) is parallel to \(\pi \).

3. Show that \(c=-1\).[1]
4. Find the exact distance between \({{l}_{c}}\) and \(\pi \).[2]
5. Point \(F\) is the foot of perpendicular from \(B\left( 6,-1,-4 \right)\) to \(\pi \) and point \(C\) is on \({{l}_{c}}\). Given that the area of triangle \(BCF\) is \(\frac{\sqrt{66}}{4}\) units\(^{2}\), find possible position vectors of \(C\).[5]