2015 AJC P1 Q8

The plane p with equation \(\mathbf{r}\cdot \left( \begin{matrix} a\\b\\ 4\end{matrix} \right)=15\)  contains point Q with coordinates \(\left( 0,5,0 \right)\). Plane p makes an acute angle of \(\alpha\) with the x-axis.

1. Find \(b\)[1]
2. Write down an equation relating \(\alpha\), \(a\) and \(b\).[1]

It is known that the angle between \(\left( \begin{matrix}a\\b\\4\end{matrix} \right)\)and the positive \(x\)-axis direction is obtuse and \(\alpha =45{}^\circ \).

3. Show that \(a=-5\).[2]

Plane \(p\), the \(x-z\) plane and the \(x-y\) plane all meet at a common point \(W\).

4. Find the position vector of  \(W\).[2]
5. The point \(M\) with coordinates \(\left( 1,0,5 \right)\) is a point on plane \(~p\). The line \(l\) is the intersection line between \(p\) and the \(x-z\) plane.
   Explain why \(M\) is on \(l\). Hence or otherwise, find the shortest distance of \(Q\) to the line \(l\).[3]