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 $k$ ].

Answer 1

$2 \cos \frac{\pi}{2 k }+2 \cos \frac{\pi}{ k }=\sqrt{3}+1$
$\cos \frac{\pi}{2 k }+\cos \frac{\pi}{ k }=\frac{\sqrt{3}+1}{2}$
Let $ \frac{\pi}{ k }=0, \cos \theta+\cos \frac{\theta}{2}=\frac{\sqrt{3}+1}{2} $
$\Rightarrow 2 \cos ^{2} \frac{\theta}{2}-1+\cos \frac{\theta}{2}=\frac{\sqrt{3}+1}{2}$
$\cos \frac{\theta}{2}=t $
$2 t^{2}+t-\frac{\sqrt{3}+3}{2}=0$
$t =\frac{-1 \pm \sqrt{1+4(3+\sqrt{3})}}{4}$
$=\frac{-1 \pm(2 \sqrt{3}+1)}{4} =\frac{-2-2 \sqrt{3}}{4}, \frac{\sqrt{3}}{2} $
$\because t \in[-1,1], \cos \frac{\theta}{2}=\frac{\sqrt{3}}{2}$
$\frac{\theta}{2}=\frac{\pi}{6} $
$\Rightarrow k =3$