whats the image of this circle under the reciprocal mapping?

Question

I need to find the image of $D(a,|a|)$ $a \in \mathbb{R}-\{0\}$ under the transformation $f(z) = 1/z$.

I was trying write $z = |a|e^{i\theta} + a$ and then $e^{i\theta}=\cos(\theta) + i \sin(\theta)$ and see what kind of line I get, but I get to nowhere.