Question

Two parallel chords of a circle of radius 2 units are at a distance $\sqrt{3}+1$ units 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] (where [k] denotes the largest integer less than or equal to k) is ___

Answer 1

Let $\theta =\frac{\pi }{2k}\Rightarrow cos\theta =\frac{x}{2}\Rightarrow cos2\theta =\frac{\sqrt{3}+1-x}{2}\Rightarrow 2co{s}^2\theta -1=\frac{\sqrt{3}+1-x}{2}\Rightarrow 2\left(\frac{x^2}{4}\right)-1=\frac{\sqrt{3}+1-x}{2}\Rightarrow x^2+x-3-\sqrt{3}=0\Rightarrow x=\frac{-1\pm \sqrt{1+12+4\sqrt{3}}}{2}...\left(Rootofquadraticequation\right)=\frac{-1\pm \sqrt{13+4\sqrt{3}}}{2}=\frac{-1+2\sqrt{3}+1}{2}=\sqrt{3}\therefore cos\theta =\frac{\sqrt{3}}{2}\Rightarrow \theta =\frac{\pi }{6}\therefore Requiredangle=\frac{\pi }{k}=2\theta =\frac{\pi }{3}\Rightarrow k=3$