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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)

Hyperbola definition

Example 1:

Hyperbola example

Hyperbola solution

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

Hyperbola equation

Example 2:

Hyperbola example

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Foci

The foci for a horizontal oriented hyperbola are given by

Hyperbola foci

The foci for a vertical oriented hyperbola are given by

Hyperbola foci

Asymptote:

Asymptotes of a horizontal oriented hyperbola are determined by

Hyperbola asymptotes

Asymptotes of a vertically oriented hyperbola are determined by

Hyperbola asymptotes

Eccentricity:

The eccentricity is given by

Hyperbola Eccentricity

Example 3:

Consider the equation

Hyperbola Eccentricity

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

Hyperbola Eccentricity

Foci:

Hyperbola foci

Asymptotes:

Hyperbola asymptotes

Eccentricity:

Hyperbola Eccentricity

Picture:

Hyperbola Eccentricity

Hyperbola centered in (u,v):

Horizontal oriented hyperbola centered at (u, v)

Hyperbola centered

Vertical oriented hyperbola centered at (u, v)

Vertical oriented hyperbola

Foci:

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

Hyperbola foci

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

analytic geometry

Asymptote:

Asymptotes of a horizontal oriented hyperbola are determined by

Asymptotes

Asymptotes of a vertically oriented hyperbola are determined by

Asymptotes

Eccentricity:

The eccentricity is given by

Eccentricity

Parametric Equations:

Horizontal oriented hyperbola:

Parametric Equations

Vertical oriented hyperbola:

Parametric Equations