If $z=x+iy$  lies in 3rd quadrant then $\overline{z}/z$  also lies in 3rd quadrant if :

Let $z=x+iy$  where $x<0,y<0$ 

So, $\frac{\overline{z}}{z}=\frac{x-iy}{x+iy}=\frac{x^2-y^2}{x^2+y^2}-\left(\frac{2xy}{x^2+y^2}\right)i$

$\frac{\overline{z}}{z}$  lies 3rd  quadrant if $x^2-y^2<0$  and $-2xy<0$

$x<0,y<0\Rightarrow -2xy<0$

Also, $x^2-y^2<0if\left(x+y\right)\left(x-y\right)<0$

If $x-y>0$  if $\left(x+y\right)\left(x-y\right)<0$

So, $\frac{\overline{z}}{z}$ lies in $III$  quadrant if $y<x<0$