\(OAC\) is a triangle and \(B\) is a point on \(OC\).
\(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=2\mathbf b\) and \(OB:BC=2:3\).
\(X\) is the point on \(AB\) such that \(AX:XB=1:2\).
Express \(\overrightarrow{AC}\), as simply as possible, in terms of \(\mathbf a\) and \(\mathbf b\).[1]
Express \(\overrightarrow{XB}\), as simply as possible, in terms of \(\mathbf a\) and \(\mathbf b\).[2]
\(Y\) is the point on \(BC\) such that \(\overrightarrow{XY}\) is parallel to \(\overrightarrow{AC}\). Find \(\overrightarrow{XY}\), in terms of \(\mathbf a\) and \(\mathbf b\).[2]
Find the ratio of the area of triangle \(AOB\) : area of triangle \(ABC\).[1]
Find the value of \(\dfrac{\text{area of triangle}\hspace{0.5em}BXY}{\text{area of trapezium}\hspace{0.5em}AXYC}\).[1]