key term - Transverse Axis

Definition

The transverse axis is a line that passes through the center of a conic section, such as an ellipse or hyperbola, and is perpendicular to the major axis. It represents the shorter of the two principal axes of the conic section and is crucial in understanding the properties and equations of these geometric shapes.

5 Must Know Facts For Your Next Test

1. The transverse axis of an ellipse or hyperbola is the shorter of the two principal axes, and is perpendicular to the major axis.
2. The length of the transverse axis, along with the length of the major axis, determines the overall size and shape of the conic section.
3. In the equation of an ellipse, the transverse axis is represented by the variable 'b', while the major axis is represented by the variable 'a'.
4. The transverse axis is an important factor in determining the eccentricity of a conic section, which in turn affects the shape and properties of the shape.
5. The transverse axis plays a crucial role in the rotation of axes, as it defines the orientation of the conic section in the coordinate plane.

Review Questions

• Explain the relationship between the transverse axis and the major axis of an ellipse or hyperbola.
  • The transverse axis and the major axis are the two principal axes of an ellipse or hyperbola. The transverse axis is the shorter of the two, and is perpendicular to the major axis. The lengths of these two axes, along with the eccentricity of the shape, determine the overall size and shape of the conic section. The transverse axis, represented by the variable 'b' in the equation, works in conjunction with the major axis, represented by 'a', to define the key properties of the ellipse or hyperbola.
• Describe how the transverse axis is involved in the rotation of axes for conic sections.
  • The transverse axis plays a crucial role in the rotation of axes for conic sections. When the coordinate system is rotated, the transverse axis defines the new orientation of the shape in the plane. This is important because the equations for conic sections, such as the standard form, rely on the orientation of the axes relative to the shape. The transverse axis, along with the major axis, determines the angle of rotation and the new coordinates of the conic section after the axes have been transformed.
• Analyze how the transverse axis is represented in polar coordinates and its significance in that context.
  • In the representation of conic sections in polar coordinates, the transverse axis is an essential element. The polar equation of an ellipse or hyperbola involves the eccentricity of the shape, which is directly related to the ratio of the transverse axis to the major axis. Additionally, the orientation of the transverse axis determines the angle at which the conic section is positioned in the polar coordinate system. This information is crucial for understanding the properties and behavior of conic sections when expressed in polar form, as the transverse axis is a key factor in defining the shape, size, and location of the curve.

Related terms

Major Axis:

The longest diameter of an ellipse or hyperbola, passing through the center of the shape and defining its overall size and orientation.

Focal Point: The points on the major axis of an ellipse or hyperbola that are equidistant from the center and the vertices of the shape, defining its shape and eccentricity.

Eccentricity:

A measure of how much a conic section deviates from a perfect circle, with a value between 0 and 1 that determines the shape of the ellipse or hyperbola.