Use two line parameters, a division ratio and a cross-product area

With reference to an origin $O$, the points $A$, $B$, $C$ and $D$ are such that $\overrightarrow{OA}=\mathbf a$, $\overrightarrow{OB}=\mathbf b$, $\overrightarrow{OC}=3\mathbf a-\mathbf b$ and $\overrightarrow{OD}=2\mathbf a+3\mathbf b$. The lines $BC$ and $AD$ intersect at point $X$ (see diagram).

(i) Express $\overrightarrow{OX}$ in terms of $\mathbf a$ and $\mathbf b$.

(ii) The point $Y$ on $BC$ is such that $BY:YC=2:11$. Show that the points $O$, $Y$ and $D$ are collinear.

(iii) Find the ratio of the area of triangle $OAB$ to the area of triangle $BCD$.