If z=x+iy  lies in 3rd quadrant then z¯/z  also lies in 3rd quadrant if :

Let z=x+iy  where x<0,y<0 So, z¯z=x−iyx+iy=x2−y2x2+y2−(2xyx2+y2)iz¯z  lies 3rd  quadrant if x2−y2<0  and −2xy<0x<0,y<0⇒−2xy<0Also, x2−y2<0  if  (x+y)(x−y)<0If x−y>0  if (x+y)(x−y)<0So, z¯z lies in III  quadrant if y