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Properties of Conjugate of a Complex Number

Consider az...

Question

Consider $a z^{2} + b z + c = 0$ . where $a, b, c \in R$ and $4 a c > b^{2}$
If $z_{1}$ and $z_{2}$ are the roots of the equation given above, then which one of the following complex numbers is purely real ?

z_{1} {¯ z}_{2}

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{¯ z}_{1} z_{2}

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z_{1} - z_{2}

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(z_{1} - z_{2}) i

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Solution

The correct option is D $(z_{1} - z_{2}) i$
Since, $Z_{1}$ and $Z_{2}$ are conjugated pairs,
let $Z_{1} = a + i b$
and $Z_{2} = a - i b$
So, $Z_{1}_{2} = (a + i b) (a - i b) = a^{2} - b^{2} + 2 a i b$
$_{1} Z_{2} = (a - i b) (a - i b) = a^{2} - b^{2} - 2 a i b$
$Z_{1} - Z_{2} = 2 a i b$
$(Z_{1} - Z_{2}) i = (2 a i b) i = - 2 a b = Purely real$

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