Let z and w  be two complex numbers such that w=zz¯−2z+2,  and z+iz-3i=1 Rew  has minimum value. Then the minimum value of n∈N  for which wn  is real, is equal to

z=x+iy⇒w=zz¯−2z+2=x2+y2−2x−2iy+2⇒Rew=x2−2x+2+y2=x−12+1+y2                                ......(1)Alsoz+i=z−3i⇒y=1                                                                         ......(2)From (1) and (2) Since real part of w is minimum, we must have x=1x=1,y=1⇒w=2-2i∴wn=2n1-in  which is real when n=4