According to given information |z−2i|=2 is a circle having centre is (0,2) and radius is 2.
If the area of △ABC is maximum the triangle must be equilateral triangle and point M is the mid point of BC, where M is the foot of perpendicular of point A(z1=2+2i) on BC. ∴ Sum of the imaginary parts of z2 and z3 is twice of the imaginary part of M=2×2=4.