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 $k$ ] [2010]

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 $k$ ] [2010]

Smart Achievers · Accepted Answer

(3)

From the figure,

$\frac{π}{k}$

$\Rightarrow 2 \times 2 \cos^{2} \frac{π}{2 k} + 2 \cos \frac{π}{2 k} - 2 = \sqrt{3} + 1$

$\Rightarrow 4 \cos^{2} \frac{π}{2 k} + 2 \cos \frac{π}{2 k} - (3 + \sqrt{3}) = 0$

$\Rightarrow \cos \frac{π}{2 k} = \frac{- 2 \pm \sqrt{4 + 16 (3 + \sqrt{3})}}{8}$ $= \frac{- 1 \pm \sqrt{13 + 4 \sqrt{3}}}{4}$

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

$As \frac{π}{2 k} is an acute angle, \cos \frac{π}{2 k} = \frac{\sqrt{3}}{2} = \cos \frac{π}{6}$ $\Rightarrow k = 3$

Solution

Q.
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 $k$ ] [2010]

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Solution

Q. Two parallel chords of a circle of radius 2 are at a distance 3+1 apart. If the 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] [2010]

Ans. (3) From the figure, 2cosπk+2cosπ2k=3+1 ⇒2×2cos2π2k+2cosπ2k-2=3+1 ⇒4cos2π2k+2cosπ2k-(3+3)=0 ⇒cosπ2k=-2±4+16(3+3)8=-1±13+434 =-1±(23+1)4=32 or -(3+12) As π2k is an acute angle, cosπ2k=32=cosπ6⇒k=3

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Q.
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 $k$ ] [2010]