## Parts of a Hyperbola

The hyperbola has certain notable features:

- The transverse axis, or major axis, is a line that passes through \(C\) and the vertices. It has a length of \(2a\).
- The conjugate axis, or minor axis, is a line that passes through \(C\) and is perpendicular to the transverse axis; It has a length of \(2b\).
- Asymptotes are lines that continually approach the hyperbola but will never meet. These are the continuous diagonals of the central rectangle.
- The central rectangle is a figure with dimensions \(2a\) and \(2b\); it is the basis for drawing the hyperbola.
- Center \(C\) is the intersection point of the transverse and conjugate axes with coordinate \((h, k)\).
- Vertex \(V\) are the endpoints of the transverse axis.
- Co-Vertex, \(V_c\) are the endpoints of the conjugate axis.
- Foci \(F\) are the two distinct fixed points that are the basis for locus definition.
- The distance \(a\), semi-major distance, is the halfway distance between vertices. It is the distance between \(C\) and \(V\).
- The distance \(b\), semi-minor distance, is the halfway distance between co-vertices. It is the distance between \(C\) and \(V_c\).
- The distance \(c\), linear eccentricity, is the distance between \(C\) and \(F\).

Distances \(a\), \(b\), and \(c\) define the shape of the hyperbola. There is a relationship between these distances:

\(c^2=a^2+b^2\)

To derive this equation, position a point \(P\) above a \(F\) on the hyperbola,

- When positioning \(P\) above \(F\), we created a right triangle using the loci definition.
- Apply the Pythagorean theorem to get the equation above.

This equation adds the squares of a and b instead of taking the difference as indicated in the ellipse relationship.

## How Open is the Hyperbola?

Sometimes we wonder "how pointed a hyperbola is." We take the ratio of distances \(c\) and \(a\) – known as eccentricity to measure this.

Eccentricity \(e\) must be greater than 1 for a hyperbola. Furthermore:

- If \(e\) nearly equals 1, it will look more pointed
- If \(e\) is farther from 1, It will look more flat

## Summary

The hyperbola is a set of all points in which the absolute value of the difference of its distances from two unique points (foci) is constant.

The hyperbola has certain notable parts: transverse axis, conjugate axis, asymptotes, central rectangle, center, vertex, covertex, focus, and distances a, b, and c.

Distances \(a\), \(b\), and \(c\) define the shape of the hyperbola. There is a relationship between these distances: \(c^2=a^2+b^2\)

Eccentricity \(e\) must be greater than 1 for a hyperbola. It will look more pointed if \(e\) nearly equals 1. It is flatter if \(e\) is farther from 1.