Let A = z ∈ C : z + 1 z - 1 < 1 and B = z ∈ C : arg z - 1 z + 1 = 2 π 3 . Then A &#8745…

Let A=z∈C:z+1z-1<1 and B=z∈C:argz-1z+1=2π3. Then A∩B is
  1. a portion of a circle centred at 0,-13 that lies in the second and third quadrants only
  2. a portion of a circle centred at 0,-13 that lies in the second quadrant only
  3. an empty set
  4. a portion of a circle of radius 23 that lies in the third quadrant only

Solution

Set A is z∈C:z+1z-1<1

⇒z¯+1z¯-1<1

⇒z¯+1<z¯-1

⇒x+12+y2<x-12+y2

⇒ x<0

Set B is z∈C:argz-1z+1=2π3

⇒argz-1z+1=2π3

⇒tan-1yx-1-tan-1yx+1=2π3

⇒x2+y2+2y3-1=0

⇒  Centre 0,-13

Hence, A∩B will represent a portion of a circle centred at 0,-13 that lies in the second quadrant only

Asked in: JEE Main 2022 (26 Jun Shift 1)

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