Consider the fractional linear map $\Phi(z)=(az+b/cz+d)$ where $a$, $b$, $c$ $d$ are real numbers with $ad-bc=1$. Suppose in addition that $|a+d| > 2$ a) show that if $c \ne 0$, there exists exactly $2$ points $x\in\Bbb R$ such that $\Phi (x)=x$. Hint: quadratic formula b) use part a) to show that there is a unique complete geodesic $g$ in $H^2$ such that that $\Phi (g)=g$.
I have solved part a) but I don't know where to begin or how to do part b)
A geodesic is determined by two points, in particular its two ideal points. If a geodesic is fixed by an isometry, its set of endpoints $E$ is sent to itself. The endpoints are either point-wise fixed, or flipped. In the first case, you know exactly what the endpoints are. As for the second, you should see if it can occur under your hypotheses.