This question is a follow-up question to half of a hyperbola.
When constructing a hyperbola in TikZ, how can I specify the eccentricity to be
Two quantities out of a, b, and e (the eccentricity) determine the hyperbola. You can, for example, define e(>1) and a(>0), from which you can then derive b. See below.
EDIT: see this link, which explains my parameterisation.