2023 JPJC Promo Q11

The plane \({{\Pi }_{1}}\) has equation \(\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} \alpha \\ \beta \\ 0 \\ \end{matrix} \\ \end{matrix} \right)=30\), where \(\alpha \) and \(\beta \) are positive constants, and contains the point \(A\) with coordinates \(\left( 0,\ 10,\ 5 \right)\).

1. Given that the perpendicular distance from the origin \(O\) to the plane \({{\Pi }_{1}}\) is 6 units, find \(\alpha \) and \(\beta \).[5]

Another plane \({{\Pi }_{2}}\) has equation \(\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} 1 \\ -1 \\ -2 \\ \end{matrix} \\ \end{matrix} \right)=4\).

2. Find the acute angle between the line \(OA\) and the plane \({{\Pi }_{2}}\).[2]
3. Find a Cartesian equation of the plane \({{\Pi }_{3}}\) which contains the line \(OA\) and is perpendicular to the plane \({{\Pi }_{2}}\).[2]