2013 CJC P1 Q9

Referred to the origin \(O\), the points \(A\) and \(B\) have position vectors given respectively by \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\). \(R\) is a point that divides \(AB\) internally in the ratio \(\lambda :\mu \). It is given that \(\overrightarrow{OR}=\mathbf{r}\) and \(\frac{\mathbf{r}\cdot \mathbf{a}}{\mathbf{r}\cdot \mathbf{b}}=\frac{\mu {{\left| \mathbf{a} \right|}^{2}}+\lambda \mathbf{a}\cdot \mathbf{b}}{\lambda {{\left| \mathbf{b} \right|}^{2}}+\mu \mathbf{a}\cdot \mathbf{b}}\).

Deduce that, when \(\lambda :\mu =\left| \mathbf{a} \right|:\left| \mathbf{b} \right|\), \(\frac{\mathbf{r}\cdot \mathbf{a}}{\mathbf{r}\cdot \mathbf{b}}=\frac{\left| \mathbf{a} \right|}{\left| \mathbf{b} \right|}\).[2]

Hence, show that the line \(OR\) bisects the angle \(AOB\).[2]