Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac{ ext { length of } \operatorname{arc} A B}{ ext { length of } \operatorname{arc} B C}=\frac{1}{5}$, and $\overrightarrow{O C}=\alpha \overrightarrow{O…

Question

Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac{	ext { length of } \operatorname{arc} A B}{	ext { length of } \operatorname{arc} B C}=\frac{1}{5}$, and $\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}$, then $\alpha+\sqrt{2}(\sqrt{3}-1) \beta$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \ldots . .(1) \\ & \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}} \\ & 0=\alpha+\beta \cos 15^{\circ} \ldots .(2) \\ & (1) \Rightarrow \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\beta \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{b}} \\ & \Rightarrow \cos 75^{\circ}=\alpha \cos 15^{\circ}+\beta \ldots .(3) \\ & (2) \&(3) \Rightarrow \cos 75^{\circ}=-\beta \cos ^2 15^{\circ}+\beta\end{aligned}$
$\beta=\frac{\cos 75^{\circ}}{\sin ^2 15^{\circ}}=\frac{1}{\sin 15^{\circ}}=\frac{2 \sqrt{2}}{\sqrt{3}-1}$
(2) $\Rightarrow \alpha=\frac{-\cos 15^{\circ}}{\sin 15^{\circ}}=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)}$
$\therefore \overrightarrow{\mathrm{c}}=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)} \overrightarrow{\mathrm{a}}+\left(\frac{2 \sqrt{2}}{\sqrt{3}-1}\right) \overrightarrow{\mathrm{b}}$
Now
$\begin{aligned}
& \alpha+\sqrt{2}(\sqrt{3}-1) \beta=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)}+\frac{\sqrt{2}(\sqrt{3}-1) \cdot 2 \sqrt{2}}{\sqrt{3}-1} \\ & =\frac{-(\sqrt{3}+1)^2}{2}+4 \\ & =\frac{-3-1-2 \sqrt{3}+8}{2} \\ & =2-\sqrt{3}
\end{aligned}$ ^

Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$,…

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