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z₁=a+i b and z₂=c+i d are two complex numbers (a, b, c, d R) such that |z₁ |= |z₂ |=1 and Im (z₁ z ₂ )=0 . If w₁=a+i c and w₂=b+i d , then -

Options

  1. AIm (w₁ w ₂ )=0
  2. BIm ( w ₂ w ₁ )=0
  3. CIm ( w₁ w₂ )=0
  4. DRe ( w₁ w ₂ )=0

Correct answer

D. Re ( w₁ w ₂ )=0

Step-by-step solution

Let z₁= ₁+i ₁ and z₂= ₂+i ₂ ( |z₁ |= |z₂ |=1 ) Now z₁ z ₂= ( ₁- ₂ )+i ( ₁- ₂ ) Im (z₁ z ₂ )=0 ₁- ₂= n , n I Now w ₁= ₁+ i ₂, w ₂= ₁+ i ₂ (A) w ₁ w ₂= ₁ ₁+ ₂ ₂+ i ( ₁- ₂ ) Im (w₁ w ₂ )=0 (B) w ₂ w ₁= ₁ ₁+ ₂ ₂+ i ( ₁- ₂ ) Im (w₂ w ₁ )=0 (C) w ₁ w ₂ = ₁+ i ₂ ₁+ i ₂ ₁- i ₂ ₁- i ₂ = ₁ ₁+ ₂ ₂+i ( ₁- ₂ . ^2 ₁+ ^2 ₂ Im ( w ₁ w ₂ )=0 (D) Similarly Re ( w ₁ w ₂ )=0

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