If z1 and z2 represent adjacent vertices of a regular polygon of n sides and if Im(z1)Re(z2)=2−1, thenn is equal to:

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If $z_{1}$ and $z_{2}$ represent adjacent vertices of a regular polygon of $n$ sides and if $\frac{Im (z_{1})}{Re (z_{2})} = \sqrt{2} - 1,$ then $n$ is equal to:

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answer is A.

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Detailed Solution

Let $z_{1} = r e^{i θ},$ then ${\bar{z}}_{1} = r e^{- i θ}$

$\frac{z_{1}}{{\bar{z}}_{1}} = e^{2 i θ} = e^{2 π i / n} \Rightarrow θ = π / n$

Also $(Im z_{1} / Re z_{1}) = \tan θ = \tan π / n = \sqrt{2} - 1 = \tan π / 8$

hence $n = 8$

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