Let z=x+i y be a complex number (x, y R) . Let A and B be two sets such that A= z:|z| 2 and B= z:(z+2 y)+ z 4 the area of region A B is

Step-by-step solution

aligned & Let Z=x+i y, x, y R & A= z:|z| 2 and B= z:(z+2 y)+ z 4 & |z| 2 and z+2 y+ z 4 aligned aligned & x^2+y^2 2 and x+i y+2 y+x-i y 4 & x^2+y^2 4 and 2 x+2 y 4 & x+y 2 aligned Point of intersection aligned x^2+y^2=4 and x+y & =2 y=2-x x^2+(2-x)^2 & =4 x^2+4-4 x+x^2 & =4 2 x^2-4 x & =0 2 x(x-2)=0 x=0, x & =2 y=2, y=0 aligned A B= Shaded Area aligned = & ₀^2 y_ circle -y_ line = ₀^2 4-x^2 -(2-x) d x = & [ x 2 4-x^2 + 4 2 ⁻¹ x 2 -2 x+ x^2 2 ]₀^2 & (2 ⁻¹ 1-4+ 4 2 )= -2 aligned