47. Sum and product of two complex number are real if and only if they are conjugate of each other.

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Prove that the sum and product of two complex numbers are real if and only if they are conjugate of each other.

Q. Assertion :If

z_{1} + z_{2} = a

and

z_{1} z_{2} = b

where

a = ¯ ¯ ¯ a

and

b = ¯ ¯ b

, then

a r g (z_{1} z_{2}) = 0.

Reason: The sum and product of two complex numbers are real if and only if they are conjugate of each other.

Q. Statement 1: If

z_{1} + z_{2} = a

and

z_{1} z_{2} = b

, where

a = a

and

b = ¯ ¯ b

, then

a r g (z_{1} z_{2}) = 0

.
Statement 2: The sum and product of two complex numbers are real if and only if they are conjugate of each other.

Q. Assertion :If

3 + i x^{y}

x^{2} + y + 4 i

are conjugate of each other, then

x^{2} + y^{2} = 3

. Reason: If sum & product of two complex numbers are real, then they are conjugates of each other.

If the complex number $x^{2} + y^{2} + 100 i$ and $29 - x^{2} y^{2} i$ are conjugate to each other, Find the value of $x^{3} + y^{3}$ when x and y are positive real number.

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