If z 1 and z ¯ 1 represents adjacent vertices of a regular polygon of n sides and if I m z 1 R e z 1 = 2 - 1 , then n is equal to

Question

If z 1 and z ¯ 1 represents adjacent vertices of a regular polygon of n sides and if I m z 1 R e z 1 = 2 - 1 , then n is equal to, 8, 16, 18, 24

MARKS App · Accepted Answer

Let z 1 = r e i θ then, z ¯ 1 = r e - i θ z 1 z ¯ 1 = e 2 i θ = e 2 π i / n ⇒ θ = π n Also, I m z 1 R e z 1 = tan θ = tan π n = 2 - 1 ⇒ tan π 8 = 2 - 1 ∴ n = 8

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