Let A = z C : Re ((1+i)z) Im ((1-i)z) and B = z C : |z - (1+i)| 2 . The area of the region represented by A B in the complex plane is

Step-by-step solution

Let z = x + iy . For the set A , we evaluate the given condition: (1+i)z = (1+i)(x+iy) = (x-y) + i(x+y) Re ((1+i)z) = x-y (1-i)z = (1-i)(x+iy) = (x+y) + i(y-x) Im ((1-i)z) = y-x The inequality for A is Re ((1+i)z) Im ((1-i)z) , which gives: x-y y-x 2x 2y y x This represents the half-plane above and including the line y = x . For the set B , the condition is |z - (1+i)| 2 . This represents a solid circular disk centered at (1, 1) with radius r = 2 . Notice that the center of the circle (1, 1) lies exactly on the bou