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Question

If z1 and ¯¯¯¯¯z1 represent adjacent vertices of a regular polygon of n sides and if Im(z1)Re(z1)=21, then n is equal to

A
4
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B
8
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C
16
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D
None of these
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Solution

The correct option is B 8
Since z1 and ¯¯¯z1 are the adjacent vertices of a regular polygon of n sides,
we have, z10¯¯¯z1=2πn and, |z1|=¯¯¯z1
Thus, z1=¯¯¯z1e2πin
Let z1=r(cosθ+isinθ)=reiθ
¯¯¯z1=reiθ
Since z1=¯¯¯z1e2πin
reiθ=reiθe2πin=re2πiniθ
θ=2πnθ or θ=πn
Therefore, z1=r(cosθ+isinθ)=r[cosπn+isinπn]
Now, Im(z1)Re(z1)=21rsin(πn)rcos(πn)=21
tanπn=21=tanπ8
n=8

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