Focus

Definition

A focus (plural: foci) is a point used to define and describe conic sections such as ellipses, parabolas, and hyperbolas. In these shapes, distances to the focus have special geometric properties.

5 Must Know Facts For Your Next Test

1. An ellipse has two foci, and the sum of the distances from any point on the ellipse to both foci is constant.
2. A parabola has one focus, and every point on the parabola is equidistant from the focus and a fixed line called the directrix.
3. In polar coordinates, conic sections can be described using their distance to a focus and an eccentricity parameter.
4. The equation of an ellipse in standard form $(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1)$ includes terms that are dependent on the distances to its foci.
5. For hyperbolas, there are two foci, and the difference in distances from any point on the hyperbola to each focus is constant.

Review Questions

• How many foci does an ellipse have and how do they relate to points on it?
• What is unique about the distance of any point on a parabola relative to its focus?
• Explain how you can describe conic sections in polar coordinates using a focus.

Related terms

Ellipse:

A conic section formed by slicing through a cone with a plane that intersects both halves of the cone. It has two foci.

Parabola:

A conic section created by intersecting a cone with a plane parallel to its side. It has one focus.

Conic Sections:

$\text{Curves obtained by slicing through cones at different angles: circles, ellipses, parabolas,}$ $\text{and hyperbolas.}$