Question

If z is a complex number such that $z^2$ = ($\overline{z}{)}^2$,then

• A. z is purely real
• B. z is purely imaginary
• C. Either z is purely real or purely imaginary
• D. None of these

Answer 1

The correct option is C

Either z is purely real or purely imaginary

Let z = x+iy, then its coonjucate $\overline{z}$ = x-iy

Given that $z^2=(\overline{z}{)}^2$

⇒ $x^2$ - $y^2$ + 2ixy = $x^2$ - $y^2$ - 2ixy ⇒ 4ixy = 0

if x ≠ 0 then y = 0 and if y ≠ 0 then x = 0