Let $\omega=0.5+0.1i$ and $r_0=\sqrt{1.5^2+0.1^2}$. Find the image of the curve $|z-\omega|=r_0$ under the transformation $f(z)=z+1/z$
Attempt: If $z=re^{i\theta}$ then $$f(z)=z+1/z=(r+1/r)\cos\theta+(r-1/r)i\sin\theta$$. We try to understand the image of the set $\{r_0e^{i\theta}+\omega: \theta\in \Bbb{R}\}$. So for any $\theta\in \Bbb{R}$ we have $$z=r_0e^{i\theta}+\omega=r_0 \cos\theta+0.5+(r_0\sin\theta+0.1)i$$ but it seems complicated to write this number in polar coordinates. Could anyone give a hint? Thanks!