Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the centre, angles of $\frac{\pi}{k}$ and $\frac{2 \pi}{k}$, where $k0$, then the value of $[k]$ is [Note : $[k]$ denotes the largest integer less than or equal to $k$ ]

Question

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the centre, angles of $\frac{\pi}{k}$ and $\frac{2 \pi}{k}$, where $k>0$, then the value of $[k]$ is [Note : $[k]$ denotes the largest integer less than or equal to $k$ ]

Aicharya · Accepted Answer

Let $	heta=\frac{\pi}{2 k}$ $ \begin{aligned} & \cos 	heta=\frac{x}{2} \ & \Rightarrow \quad \cos 2 	heta=\frac{\sqrt{3}+1-x}{2} \ & \Rightarrow 2 \cos ^2 	heta-1=\frac{\sqrt{3}+1-x}{2} \ & \Rightarrow 2\left(\frac{x^2}{4}ight)-1=\frac{\sqrt{3}+1-x}{2} \ & \Rightarrow \quad x^2+x-3-\sqrt{3}=0 \ & \Rightarrow \quad x=\frac{-1 \pm \sqrt{1+12+4 \sqrt{3}}}{2} \ & =\frac{-1 \pm \sqrt{13+4 \sqrt{3}}}{2} \ & =\frac{-1+2 \sqrt{3}+1}{2}=\sqrt{3} \ & 	herefore \quad \cos 	heta=\frac{\sqrt{3}}{2} \Rightarrow 	heta=\frac{\pi}{6} \ & 	herefore \quad 	ext { Required angle }=\frac{\pi}{k}=2 	heta=\frac{\pi}{3} \ & \Rightarrow \quad k=3 \ & \end{aligned} $

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at…

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