Let `z=x+i y` be a complex number, where `xa n dy` are real numbers. Let `Aa n dB` be the sets defined by `A={z :|z|lt=2}a n dB={z :(1-i)z+(1+i) z geq4}` . Find the area of region `AuuBdot`

Explanation

`z= x +iy`
`A={z: |z|le2}`
`rArr sqrt(x^(2) +y^(2)le 2)`
or ` x^(2) + y^(2)le 4`
`rArr ` z lies on or inside the cirle `x^(2) + y^(2) = 4`
`B = {z:(1-i)z+(1+i)barz ge 4}`
`rArr (1-i) (x+iy) +(1+i) (x-iy) ge 4 `
`rArr x + iy- y + x -iy + ix + y ge 4`
`rArr x +y ge 2`
Area of region `A nn B` is the shaded region shown in the figure.
Area `= (pi(2)^(2))/(4) -(1)/(2) xx 2 xx 2 = pi-2`