Question

The number of complex numbers $z$ satisfies $Re\left(z^2\right)=0,\left|z\right|=\sqrt{3}.$

Answer 1

$z=x+iy$
or $z^2=x^2-y^2+2ixy$
$⟹Re\left(z^2\right)=x^2-y^2$
Also, $\left|z\right|=\sqrt{x^2+y^2}$
$⟹x^2-y^2=0,x^2+y^2=3$
$⟹x^2=y^2=\frac{3}{2}$
$⟹x=\pm \sqrt{\frac{3}{2}},y=\pm \sqrt{\frac{3}{2}}$
$⟹z=\pm \sqrt{\frac{3}{2}}\pm \sqrt{\frac{3}{2}}i$
Thus, there are four complex numbers.
Ans: 4