The number of points in the complex plane that satisfying the conditions |z−2|=2 , z(1−i)+z¯(1+i)=4 , (where i=−1 ) is:

∵       Centre of the circle |z−2|=2 is   ( 2,0) and it lies on  z(1−i)+z¯(1+i)=4 .Hence, given line z(1−i)+z¯(1+i)=4  pass through the center of the circle, i.e., intersect at two points.∴   Number of solutions =2.