Question

Two parallel chords of a circle of radius $2$ are at distance $\sqrt{3}+1$ apart. If the chords subtend at the centre, angles of $\pi/k$ 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 $k$]

Answer 1

Let $ \theta=\frac{\pi}{2k} \Rightarrow cos \theta=\frac{x}{2}$

$\Rightarrow \cos 2 \theta=\frac{\sqrt{3}+1-x}{2}$
$\Rightarrow 2 \cos^2 \theta-1=\frac{\sqrt{3}+1-x}{2}$
$\Rightarrow 2\Big(\frac{x^2}{4}\Big)-1=\frac{\sqrt{3}+1-x}{2}$
$\Rightarrow x^2+ x - 3-\sqrt{3}=0$
$\Rightarrow x=\frac{-1\pm\sqrt{1+12+4\sqrt{3}}}{2}$
$=\frac{-1\pm\sqrt{13+4\sqrt{3}}}{2}$
$=\frac{-1+2\sqrt{3}+1}{2}=\sqrt{3}$
$\therefore \cos \theta=\frac{\sqrt{3}}{2} $
$\therefore$ Required angle $=\frac{\pi}{k}=2\theta=\frac{\pi}{3}\Rightarrow k=3$