The points $A$, $B$ and $C$ lie on the circumference of a circle with center $O$ and diamete

Question

The points $A$, $B$ and $C$ lie on the circumference of a circle with center $O$ and diameter $AC$. It is given that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. Find $\overrightarrow{BC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Hence show that $AB$ is perpendicular to $BC$.[3] The point $D$ is on $BC$ such that $BD:DC=3:2$.[3] Show that the area of triangle $OCD$ can be written as $k\left| \mathbf{a}\times \mathbf{b} \right|$ where $k$ is a real constant to be found.[3] Given that angle $AOB={{120}^{\circ }}$, find $\overrightarrow{ON}$ in terms of $\mathbf{a}$ where $N$ is the foot of perpendicular of $D$ to $AC$.[4]

Accepted Answer

(i) $\overrightarrow{BC}=-\mathbf{a}-\mathbf{b}$. (ii) Area of $OCD=\frac{1}{5}\left| \mathbf{a}	imes \mathbf{b} ight|$ where $k=\frac{1}{5}$ (iii) $\overrightarrow{ON}=-\frac{4}{5}\mathbf{a}$

2020 TJC Promo Q9

2020 TJC Promo Q9

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