Transverse Axis

Definition

The transverse axis of a conic section is the line segment that passes through the center of the conic and is perpendicular to the major axis. It represents the shorter of the two principal axes that define the shape of the conic section.

5 Must Know Facts For Your Next Test

1. The transverse axis is the shorter of the two principal axes that define the shape of a conic section.
2. For an ellipse, the transverse axis is the line segment that passes through the center and has the greatest length.
3. In a circle, the transverse axis is equal to the diameter of the circle.
4. The transverse axis is perpendicular to the major axis of a conic section.
5. The length of the transverse axis, along with the length of the major axis, determines the eccentricity of a conic section.

Review Questions

• Explain the relationship between the transverse axis and the major axis of a conic section.
  • The transverse axis and the major axis are the two principal axes that define the shape of a conic section. The transverse axis is the shorter of the two, passing through the center of the conic section and being perpendicular to the major axis. The length of the transverse axis, along with the length of the major axis, determines the eccentricity of the conic section, which is a measure of how elongated or flattened the shape is.
• Describe how the transverse axis differs between the various types of conic sections.
  • In a circle, the transverse axis is equal to the diameter of the circle, as the circle has no major axis and the two principal axes are of equal length. In an ellipse, the transverse axis is the longer of the two principal axes. In a parabola, the transverse axis is a single point at the vertex, while in a hyperbola, the transverse axis is the line segment that passes through the center and has the shortest length.
• Analyze the role of the transverse axis in determining the eccentricity of a conic section.
  • $$\text{Eccentricity} = \sqrt{1 - \left(\frac{\text{Transverse Axis}}{\text{Major Axis}}\right)^2}$$ The eccentricity of a conic section is a measure of how elongated or flattened the shape is, and it is directly related to the lengths of the transverse axis and the major axis. As the ratio of the transverse axis to the major axis decreases, the eccentricity increases, indicating a more elongated or flattened shape. This relationship is captured in the formula for eccentricity, which shows that the transverse axis plays a crucial role in determining the overall shape and properties of a conic section.

Related terms

Major Axis: The longest diameter of an ellipse or the line segment that passes through the center of a conic section and has the greatest length.

Minor Axis: The shortest diameter of an ellipse or the line segment that passes through the center of a conic section and is perpendicular to the major axis.

Conic Section: A curve that is the intersection of a plane and a cone, including the circle, ellipse, parabola, and hyperbola.