2016/069) Given $z'=1+i-\frac{2}{z}$. Let $z=x+iy$, prove that if $z'$ is a pure imaginary, then M moves on a circle

we have
$z'=1+i-\frac{2}{x+iy}$
multiply by conjugate to get
$z'= 1+i-\frac{2(x-iy)}{x^2+y^2}$
$=1- \frac{2x}{x^2+y^2} + i(1+ \frac{2y}{x^2+y^2})$
if it is imaginary real part is zero
or
$1- \frac{2x}{x^2+y^2} = 0$
or$x^2+y^2 - 2x = 0$
ort $x^2-2x+1 + y^2 =1$
or $(x-1)^2+y^2 = 1$
which is a circle with centre(1,0) radius 1