Question

# If A1,A2,...An are vertices of regulr polygen of n sides, inscribed in circle of radius r, whose cnetre is origin O, any P is any point on the arc An,A1 such that ∠POA1=θ. The value of sum of lenghts of the lines joining P to the angluar points of the polygen is

A
2r.sec π2n.sin (θ2+π2n)
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B
2r.cosecπ2n.cos(θ2+π2n)
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C
2r.cosecπ2n.cos(θ2π2n)
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D
2r.sec π2n.sin (θ2π2n)
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Solution

## The correct option is C 2r.cosecπ2n.cos(θ2−π2n)According to problem, ∠1OA2=∠A2OA3=...∠AnOA0=2πn ∠POA1=θ (Given) ∠POA2=θ+2πn ∠POA3=θ+2πn... Hence if r be the radius of the circle, we have PA1=2r sinPOA12=2r sinθ2 PA2=2r sinPOA22=2r sin(θ2+πn) Hence the required sum =2r[sinθ2+sin(θ2+πn)+sin(θ2+2πn)+...n terms] =2r[sinθ2+(n−12)πnsinnπ2n]sinπ2n =2r cosec π2nsin[π2+θ2−π2n] =2r cosec π2n cos(θ2−π2n)

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