N2019 P2 Q5

With reference to the origin \(O,\) the points \(A,\,B,\,C\) and \(D\) are such that \(\overrightarrow{OA}=\mathbf{a}\mathrm{,}\) \(\overrightarrow{OB}=\mathbf{b}\mathrm{,}\) \(\overrightarrow{OC}=2\mathbf{a}+4\mathbf{b}\) and \(\overrightarrow{OD}=\mathbf{b}+5\mathbf{a}\mathrm{.}\) The lines \(BD\) and \(AC\) cross at \(X\)(see diagram).

1. Express \(\overrightarrow{OX}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\mathrm{.}\)[4]

The point \(Y\) lies on \(CD\) and is such that the points \(O,\,X\) and \(Y\) are collinear.

2. Express \(\overrightarrow{OY}\) in terms of \(\mathbf{a}\)and \(\mathbf{b}\)and find the ratio \(OX:OY.\)[6]