Focus

Definition

In conic sections, the focus is a fixed point used to define and construct the curve. Each type of conic section (ellipse, parabola, hyperbola) has its own specific properties related to its focus.

5 Must Know Facts For Your Next Test

1. An ellipse has two foci; any point on the ellipse has a constant total distance to these two foci.
2. A parabola has one focus, and every point on the parabola is equidistant from this focus and a corresponding directrix.
3. A hyperbola has two foci; the difference in distances from any point on the hyperbola to these two foci is constant.
4. The coordinates of the focus (or foci) can be derived using standard equations for each conic section.
5. In polar coordinates, conic sections can be described using equations that highlight their relationship to their foci.

Review Questions

• How many foci does an ellipse have and what is their significance?
• Describe how a parabola's focus relates to any point on the curve.
• What is unique about the distances from points on a hyperbola to its two foci?

Related terms

Directrix:

A fixed line used in conjunction with a focus to define a conic section.

Eccentricity:

A parameter that determines the shape of a conic section; it describes how much the conic section deviates from being circular.

$D^2 = r_1 - r_2$: $r_1$ and $r_2$ represent distances from any point on a hyperbola to its two foci, where $D$ is constant.