Let A = z C : 0 ( z - 1 z + 1 ) 2 and B = z C : |z| 3 . The area of the region represented by A B is

Step-by-step solution

Let w = z - 1 z + 1 . Substituting z = x + iy , we get: w = (x - 1) + iy (x + 1) + iy = ((x - 1) + iy)((x + 1) - iy) (x + 1)^2 + y^2 w = x^2 + y^2 - 1 + i(2y) (x + 1)^2 + y^2 The condition 0 (w) 2 implies that both the real and imaginary parts of w must be non-negative. Re (w) 0 x^2 + y^2 - 1 0 |z| 1 Im (w) 0 2y 0 y 0 Thus, set A represents the region outside or on the unit circle ( |z| 1 ) in the upper half-plane ( y 0 ). Set B represents the region on or inside the circle of radius 3 centered at the origin ( |z|