In the mathematical literature on analytical geometry, various notions of the conics' eccentricity correspond to different approaches to these curves. After making a comparative analysis of well-known textbooks to determine coincidences and differences between the various concepts of eccentricity, as well as how the authors explain the relationship between eccentricity and the appearance of the curves, we find: (a) that these notions of eccentricity are not entirely equivalent and (b) that the authors do not adequately explain the effect of the value of eccentricity on some qualitative characteristics of the curves. Due to these two situations and because textbooks are an essential source of information for mathematics teachers, we investigated with a group of twenty high school analytical geometry teachers at a Mexican university, where we explored their conceptions and knowledge about the eccentricity of the conics. Here we report the results of this research and some details of the findings (a) and (b) obtained from the comparative analysis of textbooks.