Let z -i be any complex number such that z - i z + i is a purely imaginary number. Then z + 1 z is:

Step-by-step solution

Let z=x+i y z-i z+i is purely imaginary means its real part is zero. aligned & x+i y-i x+i y+i = x+i(y-1) x+i(y+1) x-i(y+1) x-i(y+1) &= x^2-2 i x(y+1)+x i(y-1)+y^2-1 x^2+(y+1)^2 &= x^2+y^2-1 x^2+(y+1)^2 - 2 x i x^2+(y+1)^2 aligned for pure imaginary, we have aligned & x^2+y^2-1 x^2+(y+1)^2 =0 & x^2+y^2=1 &(x+i y)(x-i y)=1 & x+i y= 1 x-i y =z aligned and 1 z =x-i y z+ 1 z =(x+i y)+(x-i y)=2 x (z+ 1 z ) is any non-zero real number