Foci

Definition

Foci are specific points used to define conic sections. For ellipses and hyperbolas, the distance from any point on the curve to the foci has unique properties that characterize these shapes.

5 Must Know Facts For Your Next Test

1. In an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant.
2. In a hyperbola, the difference of the distances from any point on one of its branches to the two foci is constant.
3. Foci lie along the major axis for ellipses and along the transverse axis for hyperbolas.
4. The distance between the center of an ellipse or hyperbola and each focus is denoted by $c$, where $c^2 = a^2 - b^2$ for ellipses and $c^2 = a^2 + b^2$ for hyperbolas.
5. For both ellipses and hyperbolas, knowing the coordinates of foci helps in graphing and understanding their geometric properties.

Review Questions

• What is unique about the sum of distances from any point on an ellipse to its two foci?
• How do you calculate $c$, the distance from the center to each focus, in a hyperbola?
• Where are the foci located in relation to an ellipse's axes?

Related terms

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

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

Major Axis: The longest diameter of an ellipse that passes through its center and both foci.