With reference to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$

Question

With reference to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are not parallel. $C$ lies on $OA$ produced with $OC:AC=3:1$ and $D$ divides $OB$ in a ratio of $2:3$. $X$ is the point of intersection of $AB$ and $CD$. Show that $\overrightarrow{OX}=\frac{9}{11}\mathbf{a}+\frac{2}{11}\mathbf{b}$.[5] Hence, find the area of the triangle $ADX$, giving your answer in the form $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be determined.[4]

Accepted Answer

(i) $\overrightarrow{OX}=\frac9{11}\mathbf a+\frac2{11}\mathbf b$. (ii) $\text{Area}(ADX)=\frac3{55}|\mathbf a\times\mathbf b|$, so $k=\frac3{55}$.

2017 AJC Promo Q9

2017 AJC Promo Q9

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