key term - Latus Rectum

Definition

The latus rectum, or parameter, of a conic section is a line segment that passes through the focus of the conic and is perpendicular to the major axis. It provides important information about the size and shape of the conic section, and its value is a key characteristic in the equations describing parabolas, ellipses, and hyperbolas.

5 Must Know Facts For Your Next Test

1. The latus rectum of a parabola is the line segment that passes through the focus and is perpendicular to the directrix, and its length is 4 times the distance between the focus and the vertex.
2. For an ellipse, the latus rectum is the line segment that passes through a focus and is perpendicular to the major axis, and its length is the minor axis of the ellipse.
3. In a hyperbola, the latus rectum is the line segment that passes through a focus and is perpendicular to the major axis, and its length is the distance between the vertices divided by the eccentricity of the hyperbola.
4. The latus rectum is an important characteristic in the equations that describe the shape and size of conic sections, as it appears in the terms that define the shape.
5. Understanding the latus rectum and how it relates to the focus, major axis, and other properties of conic sections is crucial for solving problems involving these geometric shapes.

Review Questions

• Explain how the latus rectum is defined and its relationship to the focus and major axis of a parabola.
  • The latus rectum of a parabola is the line segment that passes through the focus and is perpendicular to the directrix. Its length is 4 times the distance between the focus and the vertex of the parabola. This relationship between the latus rectum and the focus-vertex distance is an important characteristic that appears in the standard equation of a parabola, $y = ax^2 + bx + c$, where the coefficient $a$ is inversely proportional to the latus rectum.
• Describe the relationship between the latus rectum and the major and minor axes of an ellipse.
  • For an ellipse, the latus rectum is the line segment that passes through a focus and is perpendicular to the major axis. The length of the latus rectum is equal to the length of the minor axis of the ellipse. This connection between the latus rectum and the minor axis is an important geometric property that is reflected in the standard equation of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ is the length of the major axis and $b$ is the length of the minor axis.
• Explain how the latus rectum relates to the eccentricity and major axis of a hyperbola.
  • In a hyperbola, the latus rectum is the line segment that passes through a focus and is perpendicular to the major axis. The length of the latus rectum is equal to the distance between the vertices of the hyperbola divided by the eccentricity of the hyperbola. This relationship between the latus rectum, major axis, and eccentricity is a key characteristic that appears in the standard equation of a hyperbola, $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ is the length of the major axis and $e$ is the eccentricity.

Related terms

Focus:

The point within a conic section, such as a parabola, ellipse, or hyperbola, that has special geometric properties and is used in the equations describing the conic.

Major Axis:

The longest diameter of an ellipse or hyperbola, passing through the foci of the conic section.

Conic Section:

A curve that results from the intersection of a plane and a cone, including parabolas, ellipses, and hyperbolas.