Consider the Mobius transformation that maps onto the circle . Note that maps onto the circle .
Now choose an , where such that or equivalently .
Note that is the inverse of with respect to the real axis. So and must be inverses of each other with respect to . Hence or equivalently .
When . Also notice that . Therefore .
By converting into polar coordinates, the above equality yields .
So notice that only two points were chosen to map the upper-half plane onto the unit disk and certain constraints were given to a third point but no third point was chosen explicitly, therefore it makes sense that there is not a unique solution to do this.
I hope this answers your question.