Let w ⁡ = 3 + i 2 and P = w ⁡ n :  n = 1, 2, 3, … .Further H 1 = z ∈ C &#160…

Let  w⁡ = 3 + i 2  and P = w⁡ n :  n = 1, 2, 3, … .Further H 1 = z ∈ C  :  Re z > 1 2  and H 2 = z ∈ C  :  Re z < - 1 2 , where C is the set of all complex numbers. If z 1 ∈ P ∩ H 1 ,  z 2 ∈ P ∩ H 2  and O represents the origin, then ∠ z 1 O z 2 =
  1. π 2
  2. π 6
  3. 2 π 3
  4. 5 π 6

Solution


P = ω n = cos n π 6 + i  sin n π 6 ,  H 1 = Rez > 1 / 2
z 1 = P ∩ H 1 = 3 + i 2 , 3 - i 2
z 2 = P ∩ H 2 = - 1 , - 3 + i 2 , - 3 - i 2
∠ z 1 O  z 2 = 2 π 3 , 5 π 6 ,  π

Asked in: JEE Advanced 2013 (Paper 2)

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