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A complex number z is said to be unimodular if | z | = 1 . Let, z 1 and z 2 are complex numbers such that z 1 - 2z 2 2 - z 1 z ¯ 2 is unimodular and z 2 is not unimodular, then the point z 1 lies on a

Options

  1. Acircle of radius 2
  2. Bstraight line parallel to x -axis
  3. Cstraight line parallel to y -axis
  4. Dcircle of radius 2

Correct answer

D. circle of radius 2

Step-by-step solution

Given, z 1 - 2z 2 2 - z 1 z ¯ 2 is unimodular ⇒  z 1 - 2   z 2 2 - z 1 z ¯ 2 = 1 ⇒  z 1 - 2 z 2 = 2 -   z 1 z ¯ 2 Squaring both the sides, we get, z 1 - 2z 2 2 = 2 -   z 1 z ¯ 2 2 ⇒  z 1 - 2 z 2 z ¯ 1 - 2 z ¯ 2 = 2 - z 1 z ¯ 2 2 - z ¯ 1 z 2 ∵  z 2 = z  z ¯ ⇒  z 1 z ¯ 1 - 2 z 1  z ¯ 2 - 2 z ¯ 1  z 2 + 4 z 2  z ¯ 2 = 4 - 2 z ¯ 1  z 2 - 2 z 1  z &#175

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