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AP EAMCET201920 Apr 2019Evening ShiftMathematicsComplex NumberActual

For a, b, c, d R , if z₁=a+i b, z₂=c+i d are such that |z₁ |= |z₂ |=1 and Re (z₁ z ₂ )=0 , then the pair of complex numbers w₁=a+i c and w₂=b+i d satisfy

Options

  1. ARe (w₁ w ₂ )=0
  2. BRe (w₁ w ₂ )=1
  3. C|w₁ | |w₂ |
  4. D|w₁ |= |w₂ |=0

Correct answer

A. Re (w₁ w ₂ )=0

Step-by-step solution

As it is given, |z₁ |= |z₂ |=1 , so let aligned & z₁=a+i b= +i & and z₂=c+i d= +i & Now, Re (z₁ z ₂ )=a c+b d= + aligned = ( - )=0 [given] So, - = 2 or - 2 Now, for the pair of complex numbers aligned & w₁=a+i c= +i & and w₂=b+i d= +i & The Re (w₁ w ₂ )=a b+c d= + & = 1 2 [ 2 + 2 ] & = 1 2 2 ( + ) ( - )=0 [ - = 2 or - 2 ] aligned Hence, option (a) is correct.

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