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Question

If 1αk+i (αkR) are 8 vertices of a regular octagon for k=1,2,3,,8, where i=1, then the area of the octagon is

A
2
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B
22
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C
12
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D
122
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Solution

The correct option is C 12
z=1αk+i
Assuming αk=cotθ, θ(0,π)
z=sinθcosθ+isinθz=sinθ(cosθisinθ)12z=sin2θi(1cos2θ)

Let z=x+iy
Then 2x=sin2θ,2y+1=cos2θ
x2+(y+12)2=14 is the circumcircle of the octagon and circumradius, R=12


Area of the octagon is,
A=8×12R2sin(2π8)=12

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