Question

State true or false:

The region of the z-plane for which $\left|\frac{z-a}{z+\overline{a}}\right|=1(Rea\ne 0)$ is X-axis.

Answer 1

$\left|\frac{z-a}{z+\overline{a}}\right|=1\Rightarrow {|z-a|}^2={|z+\overline{a}|}^2$ If
$z=(x,y),a=(p,q)$ then
$\overline{a}=(p,q)$
$\therefore {(x-p)}^2+{(y-q)}^2={(x+p)}^2+{(y-q)}^2$
$\therefore {(x+p)}^2-{(x-p)}^2=0$or $4px=0$ $\therefore x=0$
i.e.y-axis as $p\ne 0$
This is the equation of y-axis.