Find the locus of $2/z$ given that $|z-(1+i)| = 2$ 
If complex numbers $z$ satisfy the equation $|z-(1+i)| = 2$ and $\displaystyle \omega = \frac{2}{z}$, then locus traced by $\omega$ in complex plane, is ...

My try
I want to solve it geometrically. Here $|z-(1+i)| = 2$ Represent a Circle whose center is at $(1,1)$ and Radius $=2$.
So $z$ lies on a given circle.
But I did not understand how can we find locus of $\displaystyle \omega = \frac{2}{z}$

Some ideas:
$$|z-(1+i)|=2\iff|z|\left|1-\frac{1+i}z\right|=2\iff|w|=\left|\frac2z\right|=\left|1-\frac{1+i}z\right|$$
If $\;z=x+iy\;$ , then
$$\frac{1+i}z=\frac{(1+i)\overline z}{|z|^2}=\frac{x+y}2+\frac{x-y}2i\implies 1-\frac{1+i}z=\frac{2-(x+y)}2-\frac{x-y}2i$$
But
$$2=|z-(1+i)|=|(x-1)+(y-1)i|\implies (x-1)^2+(y-1)^2=4$$
whereas
$$|w|^2=\frac14\left(4-4x-4y+x^2+2xy+y^2+x^2-2xy+y^2\right)=$$
$$=\frac12\left(x^2+y^2-2x-2y+2\right)=\frac12\left((x-1)^2+(y-1)^2\right)=2\implies$$
$$\color{red}{|w|=\sqrt2}$$
