The probability that the length of a randomly chosen chord of a circle lies between 23 and 56 of its diameter is
A
56
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B
116
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C
14
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D
512
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Solution
The correct option is C14 If l is the length of the chord, r is the distance of the mid-point of the chord from the center of the circle and a is radius of the given circle ,
then r=acosθ,l=2asinθ
Given: 232a<2asinθ<562a
⇒√116a<acosθ<√53a⇒√116a<r<√53a
∴ The given condition is satisfied if the mid-point of the chord lies within the region between the concentric circles of radius √116 and √53.
Hence the probability =( the area of the circular annulus)/( area of the given circle )