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

Answer: 3

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

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