I am able to make Mathematica plot the solution to a complex inequality as the interior or exterior of a circle : 

Let $a$ be a positive real constant, and $f(z) = \frac{z+1}{z-1}$. I want to get the equation of the image of the disc $D(a) = \{ z \in \mathbb X:\, \lvert z \rvert \le a\}$ by $f$. 

If $a<1$, its also a disc, located in the left half plane : 

    ComplexRegionPlot[Abs[(z + 1)/(z - 1)] <= 0.9, {z, 20}]

[![positive circle image][1]][1]

If $a > 1$, it's the complementary of a disc, located in the right half plane: 

    ComplexRegionPlot[Abs[(z + 1)/(z - 1)] <= 1.1, {z, 30}]

[![all the plane but the circle][2]][2]

And if $a=1$, it's the full left half plane, which in a certain sense is also a disc. 

How can I get the center and radius of this disc depending on the value of $a$ ? 


  [1]: https://i.sstatic.net/nHegX.png
  [2]: https://i.sstatic.net/Agya3.png