Foci

Definition

The foci of a conic section, such as an ellipse or hyperbola, are the two fixed points around which the curve is defined. They are the points of focus that determine the shape and properties of the conic section.

5 Must Know Facts For Your Next Test

1. The foci of an ellipse are the two points on the major axis that are equidistant from the center of the ellipse.
2. The foci of a hyperbola are the two points on the transverse axis that are the centers of the two separate branches of the hyperbola.
3. The distance between the foci of a conic section is directly related to the eccentricity of the curve, with a higher eccentricity indicating a more elongated shape.
4. The latus rectum of a conic section is perpendicular to the major axis and passes through the foci, providing another way to define the shape of the curve.
5. The directrix of a conic section, along with the foci, helps determine the equation of the curve in standard form.

Review Questions

• Explain how the foci of an ellipse are related to the shape and properties of the curve.
  • The foci of an ellipse are the two fixed points around which the curve is defined. The distance between the foci, along with the major and minor axes, determines the eccentricity of the ellipse. A smaller distance between the foci results in a more circular ellipse, while a larger distance creates a more elongated shape. The foci also play a role in defining the equation of the ellipse in standard form and the location of the latus rectum, which is perpendicular to the major axis and passes through the foci.
• Contrast the role of the foci in the definitions of an ellipse and a hyperbola.
  • While both an ellipse and a hyperbola are conic sections defined by their foci, the relationship between the foci and the shape of the curve differs. For an ellipse, the foci are located on the major axis and are equidistant from the center of the curve. This gives the ellipse a closed, oval-like shape. In contrast, the foci of a hyperbola are located on the transverse axis and are the centers of the two separate branches of the hyperbola. This results in an open, saddle-like shape for the hyperbola, rather than a closed curve.
• Analyze how the foci, eccentricity, and latus rectum of a conic section are interconnected.
  • The foci, eccentricity, and latus rectum of a conic section are all closely related concepts. The distance between the foci directly determines the eccentricity of the curve, with a larger distance resulting in a higher eccentricity and a more elongated shape. The latus rectum, which is perpendicular to the major axis and passes through the foci, provides another way to define the size and shape of the conic section. Additionally, the foci, along with the directrix, are used to derive the standard form equation of the conic section, further highlighting their central role in characterizing the properties of these curves.

Related terms

Eccentricity:

The eccentricity of a conic section is a measure of how elongated or flattened the curve is, and is related to the distance between the foci.

Latus Rectum:

The latus rectum of a conic section is the line segment that passes through the foci and is perpendicular to the major axis.

Directrix:

The directrix of a conic section is a line that is parallel to the major axis and a fixed distance from the foci, which helps define the shape of the curve.