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

Engineering

Mathematics

compound angle Formulae

Question

Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3} + 1$ apart. If the chords subtend at the center, angles of $\frac{π}{k} and \frac{2 π}{k}$ where k > 0, then the value of [k] is

[Note : [k] denotes the largest integer less than or equal to k]

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$x + y = 2 \cos \frac{π}{k} + 2 \cos \frac{π}{2 k} = \sqrt{3} + 1$ $4 a^{2} + 2 a - (\sqrt{3} + 3) = 0$

$\cos \frac{π}{k} + \cos \frac{π}{2 k} = \frac{\sqrt{3} + 1}{2}$ $a = \frac{- 2 \pm \sqrt{4 + 4 \cdot 4 \sqrt{3} (\sqrt{3} + 1)}}{8}$

$= 2 \cos^{2} \frac{π}{k} - 1 + \cos \frac{π}{2 k} = \frac{\sqrt{3} + 1}{2}$ $= \frac{- 2 \pm \sqrt{1 + 4 (3 + \sqrt{3})}}{8}$

$= 2 a^{2} + a = \frac{\sqrt{3} + 3}{2}$ $= \frac{- 1 \pm \sqrt{13 + 4 \sqrt{3}}}{4}$

$= \frac{- 1 \pm \sqrt{1 + 2 \sqrt{3}}}{4}$

$a = \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2} or a = \frac{- 2 - 2 \sqrt{3}}{4} = - (\frac{\sqrt{3} + 1}{2}) (rejected)$

$\cos \frac{π}{2 k} = \frac{\sqrt{3}}{2} \Rightarrow \frac{π}{2 k} = \frac{π}{6} \Rightarrow k = 3$