Foci

Definition

Foci are two fixed points on the interior of an ellipse or hyperbola used to define and construct these shapes. The sum of the distances from any point on an ellipse to the foci is constant, while the difference of the distances from any point on a hyperbola to the foci is constant.

5 Must Know Facts For Your Next Test

1. In an ellipse, the foci lie along the major axis.
2. The distance between each focus and the center of an ellipse is given by $c=\sqrt{a^2-b^2}$, where $a$ is the semi-major axis length and $b$ is the semi-minor axis length.
3. In a hyperbola, the foci lie along the transverse axis.
4. For a hyperbola, the distance between each focus and the center is given by $c=\sqrt{a^2+b^2}$, where $a$ is the distance from the center to a vertex and $b$ relates to its asymptotes.
5. The coordinates of foci can be used to determine other properties like eccentricity for both ellipses and hyperbolas.

Review Questions

• Where are the foci located in an ellipse?
• How do you calculate the distance from the center to each focus in a hyperbola?
• What role do foci play in defining an ellipse?

Related terms

Ellipse:

A set of all points such that the sum of their distances from two fixed points (foci) is constant.

Hyperbola:

A set of all points such that the absolute difference of their distances from two fixed points (foci) is constant.

Eccentricity:

A measure of how much an ellipse or hyperbola deviates from being circular; it depends on distances involving foci.