Let Z = x + i y is a complex number, such that x 2 + y 2 = 1 . In which of the following cases Z 1 - Z for x ≠ 1 lies in the Ι Ι n d quadrant? ∀ x , y ∈ R , i 2 = - 1

Step-by-step solution

Let, Z = cos ⁡ θ + i sin ⁡ θ Now, Z 1 - Z = cos ⁡ θ + i sin ⁡ θ 1 - cos ⁡ θ - i sin ⁡ θ = 1 2 s i n θ 2 cos ⁡ θ + i sin ⁡ θ - i 2 s i n θ 2 - i c o s θ 2 = i 2 s i n θ 2 c o s θ 2 + i s i n θ 2 = c o s θ + π 2 + i s i n θ + π 2 2 s i n θ 2 Which lies in second quadrant, if θ ∈ 0 , π ⇒ s i n θ > 0 ⇒ y > 0