Geometry of the Hyperbola

Parts of a Hyperbola

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The hyperbola has certain notable features:

1. The transverse axis, or major axis, is a line that passes through \(C\) and the vertices. It has a length of \(2a\).
2. 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 \(2b\).
3. Asymptotes are lines that continually approach the hyperbola but will never meet. These are the continuous diagonals of the central rectangle.
4. The central rectangle is a figure with dimensions \(2a\) and \(2b\); it is the basis for drawing the hyperbola.
5. Center \(C\) is the intersection point of the transverse and conjugate axes with coordinate \((h, k)\).
6. Vertex \(V\) are the endpoints of the transverse axis.
7. Co-Vertex, \(V_c\) are the endpoints of the conjugate axis.
8. Foci \(F\) are the two distinct fixed points that are the basis for locus definition.
9. The distance \(a\), semi-major distance, is the halfway distance between vertices. It is the distance between \(C\) and \(V\).
10. The distance \(b\), semi-minor distance, is the halfway distance between co-vertices. It is the distance between \(C\) and \(V_c\).
11. 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?

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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.