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Condition for Coplanarity of Four Points

An arc AC o...

Question

An arc $A C$ of a circle subtends a right angle at the centre $O$ . The point $B$ divides the arc in the ratio $1 : 2$ . If
$\to O A = \to a & \to O B = \to b,$ then the vector $\to O C$ in terms of $\to a & \to b,$ is

\sqrt{3} \to a - 2 \to b

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- \sqrt{3} \to a + 2 \to b

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2 \to a - \sqrt{3} \to b

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- 2 \to a + \sqrt{3} \to b

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Solution

The correct option is B $- \sqrt{3} \to a + 2 \to b$
Let the radius of the arc be $r$ .
Now, $B$ divides the arc in ratio $2 : 1,$ so $A B$ would subtend an angle of $60^{o}$ at the origin $O$ and similarly, BC would subtend an angle of $30^{o}$ at origin.
Let $\to a$ be $r^i . \to c = r^j$
$\to b = r$ (cos $30^{o})^i + r$ (sin $30^{o})^j = \frac{r}{2}^j + \frac{r \sqrt{3}}{2}^i$
$∴ \to b = \frac{1}{2} \to c + \frac{\sqrt{3}}{2} \to a$
$∴ 2 \to b = \sqrt{3} \to a + \to c$
$∴ \to c = - \sqrt{3} \to a + 2 \to b$

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