Question

Which of the following, locii of z on the complex plane represents a pair of straight lines?

• A. $Re\left(z^2\right)=0$
• B. $lm\left(z^2\right)=0$
• C. $|z|+z=0$
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $Re\left(z^2\right)=0$
B $lm\left(z^2\right)=0$

Consider
$z=x+iy$
$z^2=x^2-y^2+i\left(2xy\right)$
Now
$Re\left(z^2\right)=0$
$\Rightarrow (x^2-y^2)=0$
$\Rightarrow (x-y)(x+y)=0$
$x=y$ and $x=-y$ are two straight lines.
Similarly
$Im\left(z^2\right)=0$
$\Rightarrow \left(2xy\right)=0$
$x=0$ and $y=0$
That is the y and x axes respectively.