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Analytic Geometry: (lesson 3 of 3)

## Hyperbola

### Definitions:

1. A hyperbola is the locus of points where the difference in the distance to two fixed points (called the foci) is constant.

2. A hyperbola is the set of all points (x, y) in the plane the difference of whose distances from two fixed points is some constant. The two fixed points are called the foci.

A hyperbola comprises two disconnected curves called its arms or branches which separate the foci.

Hyperbola can have a vertical or horizontal orientation.

### Hyperbola centered in the origin

Standard equation of a hyperbola centered at the origin (horizontal orientation)

Example 1:

Standard equation of a hyperbola centered at the origin (vertical orientation)

Example 2:

### Foci

The foci for a horizontal oriented hyperbola are given by

The foci for a vertical oriented hyperbola are given by

### Asymptote:

Asymptotes of a horizontal oriented hyperbola are determined by

Asymptotes of a vertically oriented hyperbola are determined by

### Eccentricity:

The eccentricity is given by

Example 3:

Consider the equation

Find: a, b, foci, asymptotes, and eccentricity.

Foci:

Asymptotes:

Eccentricity:

Picture:

### Hyperbola centered in (u,v):

Horizontal oriented hyperbola centered at (u, v)

Vertical oriented hyperbola centered at (u, v)

Foci:

The foci for a horizontal oriented hyperbola centered at (u, v):

The foci for a vertical oriented hyperbola centered at (u, v):

Asymptote:

Asymptotes of a horizontal oriented hyperbola are determined by

Asymptotes of a vertically oriented hyperbola are determined by

Eccentricity:

The eccentricity is given by

### Parametric Equations:

Horizontal oriented hyperbola:

Vertical oriented hyperbola: