How Conics Govern Möbius Transformations

@article{Frantz2004HowCG,
  title={How Conics Govern M{\"o}bius Transformations},
  author={Marc Frantz},
  journal={The American Mathematical Monthly},
  year={2004},
  volume={111},
  pages={779-790}
}
where a, b, c, and d are complex constants with ad bc : 0. These are important mappings in complex analysis, and textbooks on the subject invariably give advice on how to visualize them. Perhaps the best known fact in this regard is that MObius transformations map circles to circles, provided straight lines are considered as special cases of circles. We restrict the scope of this article to true circles (not straight lines), and answer the following question: 
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