Let a= Im ( 1+z^2 2 i z ) , where z is any non-zero complex number. The set A = a:|z|=1 and z 1 is equal to:

Step-by-step solution

Let z=x+i y z^2=x^2-y^2+2 i x y Now, aligned & 1+z^2 2 i z = 1+x^2-y^2+2 i x y 2 i(x+i y) = (x^2-y^2+1 )+2 i x y 2 i x-2 y & = (x^2-y^2+1 )+2 i x y -2 y+2 i x -2 y-2 i x -2 y-2 i x & = y (x^2+y^2-1 )+x (x^2+y^2+1 ) i 2 (x^2+y^2 ) & aligned a= x (x^2+y^2+1 ) 2 (x^2+y^2 ) Since, |z|=1 x^2+y^2 =1 aligned & x^2+y^2=1 & a= x(1+1) 2 1 =x & Also z 1 x+i y 1 & A =(-1,1) aligned