Conjugate of a Hyperbola
I. Introduction to Conjugate of Hyperbola:
- The term "conjugate" in the context of hyperbolas refers to a related hyperbola that shares the same center.
- Hyperbolas come in pairs: one with a horizontal axis and the other with a vertical axis. These are considered conjugate hyperbolas.
II. Key Definitions:
- The hyperbola that is initially described in an equation is often referred to as the primary hyperbola.
- For example, or .
- The hyperbola obtained by switching the positions of and and negating one of the terms is the conjugate hyperbola.
- If the primary hyperbola is , the conjugate hyperbola is .
III. Relationship between Axes:
- The primary and conjugate hyperbolas share the same center but have their major and minor axes switched.
- If the primary hyperbola has a horizontal major axis, the conjugate hyperbola has a vertical major axis, and vice versa.
IV. Eccentricity and Shape:
- The eccentricities of the primary and conjugate hyperbolas are equal.
- If the primary hyperbola is characterized by eccentricity , the conjugate hyperbola also has eccentricity .
V. Length of Axes:
- The length of the major axis of the primary hyperbola is , while the length of the minor axis is .
- The length of the major axis of the conjugate hyperbola is , and the length of the minor axis is .
- Consider the primary hyperbola .
- The conjugate hyperbola is .
- Both hyperbolas have the same center but differ in the orientation of their major and minor axes.