Question

Integer type questions:
Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}$ + 1 apart. .If the chords subtend at the centre angle of $\frac{\pi }{k}$and$\frac{2\pi }{k}$ where k > 0. Then the value of (k) is

Answer 1

We know that distance between parallel chords is greater than radius therefore, both chords lie on opposite side of centre

$2cos\frac{\pi }{2k}+2cos\frac{\pi }{k}=\sqrt{3}+1$

If we take$\frac{\pi }{2k}=\theta$

Then $2cos\theta +2cos2\theta =\sqrt{3}+1$

$\Rightarrow 2cos\theta +2(2co{s}^2\theta -1)=\sqrt{3}+1$

$\Rightarrow 4co{s}^2\theta +2cos\theta -(3+\sqrt{3})=0$

Solving for cos$\theta$, we get

$cos\theta =\frac{-1\pm (2\sqrt{3}+1)}{4}$

$cos\lambda \frac{\pi }{2k}=\frac{\sqrt{3}}{2},\frac{-(\sqrt{3}+1)}{2}$

$\therefore$ $\frac{\pi }{2k}=\frac{\pi }{6}\Rightarrow k=3\Rightarrow \left[k\right]=3$