Question

The region of the z-plane for which $\left|\frac{z-a}{z+\overline{a}}\right|=1\left[Re\right(a)\ne 0]$ ,then real part of z is

Answer 1

$|z-a|=|z+\overline{a}|$
Now
Let $z=x+iy$ and $a=c+ib$
Then
$(x-c{)}^2+(y-b{)}^2=(x+c{)}^2+(y-b{)}^2$
$(x+c{)}^2-(x-c{)}^2=0$
$\left(2x\right)\left(2c\right)=0$
$x=0$ or $c=0$.
Since $Re\left(a\right)\ne 0$
$x=0$ Depicts the y axis.