Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 6 / b / Solution

A Möbius transformation maps generalized circles to generalized circles on the Riemann sphere. A generalized circle is an ordinary circle or a straight line together with the point at infinity. Its image is a straight line precisely when it contains infinity, equivalently when the preimage of infinity lies on the extended real axis.

If $c=0$, the pole is infinity itself and the transformation is the affine map $(a/d)z+b/d$, taking the extended real axis to a line. If $c\ne 0$, the pole is $-d/c$, which is real exactly when $d/c$ is real. Thus the full condition isEquivalently, it is $\mathrm{Im}(d\bar{c})=0$, a condition also valid when $c=0$. If it fails, the image misses infinity and is therefore an ordinary circle. The nonzero determinant ensures that none of these generalized circles collapses to a point.