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JEE Main20246 Apr 2024Evening ShiftMathematicsComplex NumberActual

If z₁, z₂ are two distinct complex number such that | z₁-2 z₂ 1 2 -z₁ z ₂ |=2 , then

Options

  1. Az₁ lies on a circle of radius 1 2 and z₂ lies on a circle of radius 1 .
  2. Bboth z₁ and z₂ lie on the same circle. both z₁ and z₂ lie on the same circle.
  3. Ceither z₁ lies on a circle of radius 1 2 or z₂ lies on a circle of radius 1 .
  4. Deither z₁ lies on a circle of radius 1 or z₂ lies on a circle of radius 1 2 .

Correct answer

D. either z₁ lies on a circle of radius 1 or z₂ lies on a circle of radius 1 2 .

Step-by-step solution

aligned & z₁-2 z₂ 1 2 -z₁ z ₂ z ₁-2 z ₂ 1 2 - z ₁ z₂ =4 & |z₁ |^2 2 z₁ z ₂-2 z ₁ z₂+4 |z₂ |^2 & =4 ( 1 4 - z ₁ z₂ 2 - z₁ z ₂ 2 + |z₁ |^2 |z₂ |^2 ) aligned aligned & (z, z ₁-1 ) (1-2 z₂ 2 z ₂ )=0 & ( |z₁ |^2-1 ) ( |2 z₂ |^2-1 )=0 aligned

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