ACJC/2021/JC1/Promo/P1/Q05

Referred to the origin $O$, points $A$ and $B$ have position vectors a and $\mathbf{b}$ respectively. The modulus of $\mathbf{a}$ is 2 and $\mathbf{b}$ is a unit vector. The angle between $\mathbf{a}$ and $\mathbf{b}$ is $60^{\circ}$. Point $C$ lies on $A B$, between $A$ and $B$, such that $A C=k C B$, where $0<k<1$.
(i) Express $\overrightarrow{O C}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.
(ii) Show that the length of projection of $\overrightarrow{O C}$ on $\overrightarrow{O A}$ is given by $\frac{k+4}{2(k+1)}$.

[3]

(iii) Find, in terms of $k$, the area of triangle $O A C$.

[3]