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Two parallel chords of a circle of radiu...

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 _____

Text Solution

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1
Let `theta=(pi)/(2k)rArr costheta=(x)/(2)`
`rArr cos2theta=(sqrt3+_1-x)/(2)`
`rArr 2cos^(2)theta-1=(sqrt3+1-x)/(2)`
`rArr 2((x^(2))/(4))-1=(sqrt3+1-x)/(2)`
`rArr x^(2)+x-3-sqrt3=0`
`rArr x= (-1pmsqrt(1+12+4sqrt3))/(2)`
`= (-1pmsqrt(1+12+4sqrt3))/(2)=(-1+2sqrt3+1)/(2)=sqrt3`
`therefore costheta=(sqrt3)/(2)rArr theta=(pi)/(6)`
`therefore " Required angle " = (pi)/(k)=2theta=(pi)/(3)`
`rArr k = 3`
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