I have been asked to find the locus of points of constant distance $d$ from the vertical axis $\{ \textrm{Re}(z)=0\}$ in the Poincaré half-plane model in hyperbolic geometry.
Here are my thoughts on this problem: I believe the geodesics in the Poincaré half plane model are such that the point on the imaginary axis of the shortest distance from the point $a+bi$ is the point $bi$ (as in the Euclidean case). We know that the distance between two points $A$ and $B$ is given by $d(A,B)=|\textrm{log}[A,B,X,Y]|$, where $X$ and $Y$ are the points of intersection between the line between $A$ and $B$, and the absolute, and $[A,B,X,Y]$ denotes the cross ratio.
My idea was to somehow combine these two facts to arrive at an equation for the locus, but I seem to be at a loss as to how to progress.
I was given the hint that I might take one point of a given distance from the imaginary axis and apply isometries of the half-plane model to generate other points of equal distance. This seems to be an approach quite different from what I initially had in mind, and I cannot see how we might arrive at a general expression by this method. We would want an isometry that preserved the imaginary axis, so this would have to be a reflection about the imaginary axis, but this just gives us the point $-a+bi$ from the point $a+bi$, and I am sure we can all agree that there needs no ghost come from the grave to tell us that these two points are of equal distance from the imaginary axis. Which isometries can we apply that preserve the imaginary axis yet generate new non-trivial points of equal distance from the axis as a given point?
All help or input would be highly appreciated.