Q.

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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a

8

b

16

c

18

d

24

answer is A.

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Detailed Solution

Let z1=reiθ,  then z¯1=re−iθ z1z¯1=e2iθ=e2πi/n⇒θ=π/n Also (Imz1/Re z1)=tanθ=tanπ/n=2−1=tanπ/8 hence n=8
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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: