2023 MI P2 Q5

The plane \(p\) has equation \(x+2y-5z=4\), and contains the line \(l\) with equation \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} 3 \\ -2 \\ b \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} a \\ 2a \\ 2 \\ \end{matrix} \\ \end{matrix} \right)\), where \(a\) and \(b\) are constants and \(\lambda \) is a parameter.

1. Find the perpendicular distance from the origin to \(p\).[2]
2. Show that \(a=2\) and \(b=-1\).[3]

Relative to \(O\), the point \(Q\) has position vector \(-4\mathbf{i}+7\mathbf{j}+3\mathbf{k}\).

3. The points \(C\) and \(D\) are on \(l\) such that they are a distance of \(\sqrt{110}\) away from \(Q\). Find the position vectors \(\overrightarrow{OC}\) and \(\overrightarrow{OD}\).[3]
4. Without performing any further calculations, explain whether the shortest distance from \(Q\) to \(l\) is greater or smaller than \(\sqrt{110}\).[1]