Question

If $z$ is a complex number such that $z^2=(\overline{z}{)}^2$, then find the location of $z$ on the Argand plane.

• A. If $x\ne 0$, then $y=0$ and if $y\ne 0$,then $x=0$.($xy=0$)
• B. both $x$ and $y$ must be zeros
• C. both $x$ and $y$ must be non zeros
• D. $x+y$=0

Answer 1

The correct option is A If $x\ne 0$, then $y=0$ and if $y\ne 0$,then $x=0$.($xy=0$)

Let, $z=x+iy⟹\overline{z}=x-iy$
Given that
$z^2=(\overline{z}{)}^2⟹x^2-y^2+2ixy=x^2-y^2-2ixy⟹4ixy=0$.
If $x\ne 0$, then $y=0$ and if $y\ne 0$,then $x=0$.
Ans: A