Two parallel chords of a circle of radius 2 are at a distance 3+1 apart. If chords subtend at the center, angles of πk and 2πk where k>0,then the value of k is[Note: k denotes the largest integer less than or equal to k]

Chord lie on opposite side of diameterLet θ1=π2k,θ2=πkcos⁡θ2=d22and cos⁡θ1=d12d1+d22=cosπ2k+cosπk3+12=cosπ2k+cosπk⇒k=3

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Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3} + 1$ apart. If 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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answer is 3.

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Detailed Solution

Chord lie on opposite side of diameter

Let $θ_{1} = \frac{π}{2 k}, θ_{2} = \frac{π}{k}$

$\begin{array}{l} \cos θ_{2} = \frac{d_{2}}{2} a n d \cos θ_{1} = \frac{d_{1}}{2} \\ \frac{d_{1} + d_{2}}{2} = \cos \frac{π}{2 k} + \cos \frac{π}{k} \\ \frac{\sqrt{3} + 1}{2} = \cos \frac{π}{2 k} + \cos \frac{π}{k} \Rightarrow k = 3 \end{array}$