Part 4/5:
Focal Length: The relationship between the semi-major axis (a), semi-minor axis (b), and the distance to the foci (c) is defined as (c^2 = a^2 + b^2).
Vertices: The vertices of the hyperbola occur at the points ((\pm a, 0)), and they represent the closest points on the hyperbola to the origin.
Asymptotes: The hyperbola is also characterized by its asymptotes, which are lines that the hyperbola approaches but never intersects. The equations of the asymptotes for the hyperbola centered at the origin are given as (y = \pm \frac{b}{a} x).
Alternate Forms
If one were to switch the roles of (x) and (y) analytically, you can arrive at a different hyperbola form involving vertical branches:
[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
]