 Solution

For an affinely parametrized geodesic $s\mapsto (x(s),y(s))$, the energy Lagrangian is $L=({\dot{x}}^2+{\dot{y}}^2)/(2y^2)$. The coordinate $x$ is cyclic, so the Euler-Lagrange equation gives $\dot{x}/y^2=c$, a constant. Parametrize a nonconstant geodesic by hyperbolic arc length; thenIf $c=0$, $x$ is constant: these are vertical lines. If $c\ne 0$, on intervals with $\dot{y}\ne 0$,Integration gives $(x-a{)}^2+y^2=c^{-2}$. The two branches join smoothly at the highest point. Conversely these curves satisfy the geodesic equations, or are images of vertical lines under half-plane Möbius transformations that are isometries. Thus the complete nonconstant geodesics arenamely vertical lines and upper semicircles orthogonal to the real axis. Their subarcs give geodesic segments.