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Properties of Conjugate of a Complex Number

If z 1, z 2, ...

Question

If $z_{1}, z_{2}, z_{3}$ are the solutions of $z^{2} + ¯ ¯ ¯ z = z,$ then $z_{1} + z_{2} + z_{3}$ is equal to
( $z$ is a complex number on the Argand plane and $i = \sqrt{- 1}$ )

A

2 + 2 i

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B

2 - 2 i

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C

0

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D

2

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Solution

The correct option is D $2$
$z^{2} + ¯ ¯ ¯ z = z$
Put $z = x + i y$
$(x + i y)^{2} + x - i y = x + i y$
$\Rightarrow x^{2} - y^{2} + i 2 x y + x - i y - x - i y = 0$
$\Rightarrow x^{2} - y^{2} + i (2 x y - 2 y) = 0 + 0 \cdot i$
Comparing real and imaginary parts, we get
$x^{2} - y^{2} = 0$ and $2 y (x - 1) = 0$
$\Rightarrow y = \pm x \dots (1)$
and $2 y (x - 1) = 0$
If $y = 0,$ then $x = 0 [From (1)]$
If $x - 1 = 0,$ then $y = \pm 1$
$∴ z_{1} = 0, z_{2} = 1 + i, z_{3} = 1 - i$
$z_{1} + z_{2} + z_{3} = 0 + 1 + i + 1 - i$
$\Rightarrow z_{1} + z_{2} + z_{3} = 2$

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Similar questions

z_{1}, z_{2}, z_{3}

are the solutions of

z^{2} + ¯ ¯ ¯ z = z

z_{1} + z_{2} + z_{3}

is equal to (where

z

is a complex number on the argand plane and

i = \sqrt{- 1})

-

z - 1, z_{2}

z_{3}

are complex numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = | (1 / z_{1}) + (1 / z_{2}) + (1 / z_{2}) | = 1

| z_{1} + z_{2} + z_{3} |

z_{1}, z_{2}, z_{3}

are three complex numbers, then

| z_{2} + z_{3} |^{2} + | z_{3} + z_{1} |^{2} + | z_{1} + z_{2} |^{2}

| z_{1} | = | z_{2} | = | z_{3} | = 1

z_{1} + z_{2} + z_{3} + = \sqrt{2} + i

, then the complex number

z_{2}_{3} + z_{3} +_{1} + z_{1} +_{2}

z_{1}, z_{2}, z_{3}

are unlmodular complex numbers then the greatest value of

{| z_{1} - z_{2} |}^{2} + {| z_{2} - z_{3} |}^{2} {+ | z_{3} - z_{1} |}^{2}

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