Two parallel chords of a circle of radius 2 are at a distance `sqrt3 + 1` apart. If the chord subtend angles `(pi)/(k) and (2pi)/(k)` at the center, where `k gt 0`, then the value of [k] is _____

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

Let AB and CD betwoparallel chords of a circle of radius 2 making angles `(pi)/(k)and (2pi)/(k)` respectively at the centre O.
We have, `LM=sqrt3+1`

`impliesOL+OM=sqrt3+1`
`2cos""(pi)/(2k)+2cos""(pi)/(k)=sqrt3+1`
`impliescos""(pi)/(2k)+cos""(pi)/(k)=(sqrt3+1)/(2)`
`implies2cos^(2)""(pi)/(2k)+cos""(pi)/(2k)-((sqrt3+3)/(2))=0`
`impliescos""(pi)/(2k)=(-1+-sqrt(1+4(sqrt3+3)))/(4)`
`impliescos""(pi)/(2k)=(-1pm(2sqrt3+1))/(4)`
`impliescos""(pi)/(2k)=(sqrt3)/(2),(-(sqrt3+1))/(2)implies(pi)/(1k)=(pi)/(6)impliesk=3,`
Hence, `[k]=3.`