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KVPY2019MathematicsComplex Number

Let z = x + i y and w = u + i v be complex numbers on the unit circle such that z 2 + w 2 = 1 . Then the number of ordered pairs z , w is

Options

  1. A0
  2. B4
  3. C8
  4. Dinfinite

Correct answer

C. 8

Step-by-step solution

It is given that complex numbers z = x + i y and w = u + i v are on the unit circle such that z 2 + w 2 = 1 . . . ( i ) So, z 2 ¯ + w 2 ¯ = 1 ⇒ z ¯ 2 + w ¯ 2 = 1 ⇒ z ¯ z z 2 + w ¯ w w 2 = 1 ⇒ 1 z 2 + 1 w 2 = 1 ⇒ z 2 + w 2 = z 2 w 2 ⇒ z 2 w 2 = 1 . . . ( ii ) ∵ The number of solution of equations x 2 + y 2 = 1 and x 2 y 2 = 1 is eight ∴ Number of ordered pair z , w is also eight.

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