Find, in terms of a and b $\overrightarrow{\text{OF}}$.

Find, in terms of a and b $\overrightarrow{\text{AF}}$.

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\lambda$;

Find an expression for $\overrightarrow{\text{OD}}$ in terms of a, b and $\mu$.

Show that $\mu =\frac{1}{13}$, and find the value of $\lambda$.

Deduce an expression for $\overrightarrow{\text{CD}}$ in terms of a and b only.

Given that area $\mathrm{\Delta }\text{OAB}=k(\text{area }\mathrm{\Delta }\text{CAD})$, find the value of $k$.

$\overrightarrow{\text{OF}}=\frac{1}{7}$b     A1

[1 mark]

$\overrightarrow{\text{AF}}=\overrightarrow{\text{OF}}-\overrightarrow{\text{OA}}$     (M1)

$=\frac{1}{7}$b – a     A1

[2 marks]

$\overrightarrow{\text{OD}}=$ a $+\lambda \left(\frac{1}{7}b-a\right)\left(=(1-\lambda )a+\frac{\lambda }{7}b\right)$     M1A1

[2 marks]

$\overrightarrow{\text{OD}}=\frac{1}{2}$ a $+\mu \left(-\frac{1}{2}a+b\right)\left(=\left(\frac{1}{2}-\frac{\mu }{2}\right)a+\mu b\right)$     M1A1

[2 marks]

equating coefficients:     M1

$\frac{\lambda }{7}=\mu ,1-\lambda =\frac{1-\mu }{2}$     A1

solving simultaneously:     M1

$\lambda =\frac{7}{13},\mu =\frac{1}{13}$     A1AG

[4 marks]

$\overrightarrow{\text{CD}}=\frac{1}{13}\overrightarrow{\text{CB}}$

$=\frac{1}{13}\left(b-\frac{1}{2}a\right)\left(=-\frac{1}{26}a+\frac{1}{13}b\right)$     M1A1

[2 marks]

METHOD 1

$\text{area }\mathrm{\Delta }\text{ACD}=\frac{1}{2}\text{CD}\times \text{AC}\times \sin \mathrm{A}\hat{\mathrm{C}}\mathrm{B}$     (M1)

$\text{area }\mathrm{\Delta }\text{ACB}=\frac{1}{2}\text{CB}\times \text{AC}\times \sin \mathrm{A}\hat{\mathrm{C}}\mathrm{B}$     (M1)

$\text{ratio }\frac{\text{area }\mathrm{\Delta }\text{ACD}}{\text{area }\mathrm{\Delta }\text{ACB}}=\frac{\text{CD}}{\text{CB}}=\frac{1}{13}$     A1

$k=\frac{\text{area }\mathrm{\Delta }\text{OAB}}{\text{area }\mathrm{\Delta }\text{CAD}}=\frac{13}{\text{area }\mathrm{\Delta }\text{CAB}}\times \text{area }\mathrm{\Delta }\text{OAB}$     (M1)

$=13\times 2=26$     A1

METHOD 2

$\text{area }\mathrm{\Delta }\text{OAB}=\frac{1}{2}\left|a\times b\right|$     A1

$\text{area }\mathrm{\Delta }\text{CAD}=\frac{1}{2}\left|\overrightarrow{\text{CA}}\times \overrightarrow{\text{CD}}\right|$ or $\frac{1}{2}\left|\overrightarrow{\text{CA}}\times \overrightarrow{\text{AD}}\right|$     M1

$=\frac{1}{2}\left|\frac{1}{2}a\times \left(-\frac{1}{26}a+\frac{1}{13}b\right)\right|$

$=\frac{1}{2}\left|\frac{1}{2}a\times \left(-\frac{1}{26}a\right)+\frac{1}{2}a\times \frac{1}{13}b\right|$     (M1)

$=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{13}\left|a\times b\right|\left(=\frac{1}{52}\left|a\times b\right|\right)$     A1

$\text{area }\mathrm{\Delta }\text{OAB}=k(\text{area }\mathrm{\Delta }\text{CAD})$

$\frac{1}{2}\left|a\times b\right|=k\frac{1}{52}\left|a\times b\right|$

$k=26$     A1

[5 marks]