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For a complex number z , the equation z 2 + p + i q z + r + i s = 0 has a real root (where p , q , r , s are non-zero real numbers and i 2 = - 1 ), then

Options

  1. Ap q r = r 2 + p 2 s
  2. Bp r s = q 2 + r 2 p
  3. Cq r s = p 2 + s 2 q
  4. Dp q s = s 2 + q 2 r

Correct answer

D. p q s = s 2 + q 2 r

Step-by-step solution

Since, the equation has a real root ∴ z = x + i y = x ⇒ x 2 + p + i q x + r + i s = 0 ∴ x 2 + p x + r = 0 and q x + s = 0 Putting x = - s q in the first equation, we get, p q s = s 2 + q 2 r which is the required condition.

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