If z is a complex number such that z2=( barz)2, then

Question

If z is a complex number such that z2=( barz)2, then (A) z is purely real (B) z is purely imaginary (C) either z is purely real or purely imaginary (D) None of these

Tardigrade Admin · Accepted Answer

Let z=x+ iy, then its conjugate barz=x-i y Given that z2=( barz)2 ⇒ x2-y2+2 i x y=x2-y2-2 i x y ⇒ 4 i x y=0 If x ≠ 0, then y=0 and if y ≠ 0, then x=0

Q. If $z$ is a complex number such that $z^{2} = (\overset{z}{ˉ})^{2}$ , then

$z$ is purely real

$z$ is purely imaginary

either $z$ is purely real or purely imaginary

None of these

Let $z = x +$ iy, then its conjugate $\overset{z}{ˉ} = x - i y$
Given that $z^{2} = (\overset{z}{ˉ})^{2}$
$\Rightarrow x^{2} - y^{2} + 2 i x y = x^{2} - y^{2} - 2 i x y$
$\Rightarrow 4 i x y = 0$
If $x \neq = 0$ , then $y = 0$ and if $y \neq = 0$ , then $x = 0$

Q. If z is a complex number such that z2=(zˉ)2, then

z is purely real

z is purely imaginary

either z is purely real or purely imaginary