key term - Semi-Minor Axis

Definition

The semi-minor axis is one of the two principal axes of an ellipse, the other being the semi-major axis. It is the shorter of the two axes and represents the distance from the center of the ellipse to the edge along the minor axis.

5 Must Know Facts For Your Next Test

1. The semi-minor axis, along with the semi-major axis, defines the size and shape of an ellipse.
2. The length of the semi-minor axis is always less than or equal to the length of the semi-major axis.
3. The ratio of the semi-minor axis to the semi-major axis is known as the aspect ratio, which determines the eccentricity of the ellipse.
4. The semi-minor axis is one of the key parameters used to describe the equation of an ellipse in standard form.
5. The semi-minor axis is an important characteristic in the study of conic sections, as it, along with the semi-major axis, determines the properties and behavior of ellipses.

Review Questions

• Explain the relationship between the semi-minor axis and the semi-major axis of an ellipse.
  • The semi-minor axis and the semi-major axis are the two principal axes that define the size and shape of an ellipse. The semi-minor axis represents the shorter of the two axes, while the semi-major axis represents the longer axis. The ratio of the semi-minor axis to the semi-major axis is known as the aspect ratio, and it determines the eccentricity of the ellipse. The semi-minor axis is always less than or equal to the length of the semi-major axis, and the two axes together are used to describe the equation of an ellipse in standard form.
• Describe the role of the semi-minor axis in the study of conic sections.
  • The semi-minor axis is a crucial characteristic in the study of conic sections, particularly ellipses. Along with the semi-major axis, the semi-minor axis defines the size and shape of an ellipse, which is one of the four main conic sections. The semi-minor axis, in combination with the semi-major axis, determines the eccentricity of the ellipse, which is a measure of how much the shape deviates from a perfect circle. Additionally, the semi-minor axis is a key parameter used in the standard form equation of an ellipse, which is essential for understanding the properties and behavior of this type of conic section.
• Analyze how changes in the semi-minor axis of an ellipse affect the overall shape and characteristics of the conic section.
  • Changing the length of the semi-minor axis of an ellipse can significantly impact the overall shape and characteristics of the conic section. As the semi-minor axis decreases relative to the semi-major axis, the ellipse becomes more elongated and eccentricity increases, approaching the shape of a parabola. Conversely, as the semi-minor axis increases, the ellipse becomes more circular, with the eccentricity decreasing towards a perfect circle. These changes in the semi-minor axis affect the equation of the ellipse, as well as its properties, such as the location of the foci and the length of the major and minor axes. Understanding the role of the semi-minor axis is crucial for analyzing and working with ellipses, which are fundamental conic sections in mathematics.

Related terms

Semi-Major Axis: The semi-major axis is the longer of the two principal axes of an ellipse. It represents the distance from the center of the ellipse to the edge along the major axis.

Eccentricity:

The eccentricity of an ellipse is a measure of how much the shape deviates from a perfect circle. It is calculated as the ratio of the distance between the foci to the length of the major axis.

Conic Sections:

Conic sections are the shapes formed by the intersection of a plane with a cone. The four main conic sections are circles, ellipses, parabolas, and hyperbolas.