2024 JPJC P1 Q6

Referred to the origin \({O}\), points \({A}\) and \({B}\) have position vectors \({\mathbf{a}}\) and \({\mathbf{b}}\) respectively such that angle \({AOB = 30^\circ}\) and \(\left| \mathbf{a} \right|=4\left| \mathbf{b} \right|\).

1. Point \({C}\) has position vector \({m\mathbf{a} + n\mathbf{b}}\), where \({m}\) and \({n}\) are positive integers. Find the area of triangle \({ABC}\) in terms of \({m}\), \({n}\) and \(\left| \mathbf{b} \right|\).[4]

Point \({D}\) is the mid-point of \({OA}\) and point \({E}\) lies on \({OB}\) such that \({OE:EB = 3:2}\).

2. Find the position vectors \({\overrightarrow{OD}}\) and \({\overrightarrow{OE}}\), giving your answers in terms of \({\mathbf{a}}\) and \({\mathbf{b}}\).[2]
3. Show that the vector equation of the line \({BD}\) can be written as \({\mathbf{r} = \lambda \mathbf{a} + (1 - 2\lambda)\mathbf{b}}\), where \({\lambda}\) is a parameter. By finding the vector equation of the line \({AE}\) in a similar form in terms of a parameter \({\mu}\), find, in terms of \({\mathbf{a}}\) and \({\mathbf{b}}\), the position vector of the point \({F}\) where the lines \({BD}\) and \({AE}\) meet.[4]