The values of 'a' and 'b' determines the 'fatness' and 'skinniness' of hyperbola
A formula: a^2 e^2 = a^2 + b^2
Horizontal hyperbola
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
a^2 e^2 = a^2 + b^2
Vertical hyperbola
\frac{y^2}{a^2}-\frac{x^2}{b^2} = 1
a^2 e^2 = a^2 + b^2
Something to help...