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:

  1. Primary Hyperbola:

    • The hyperbola that is initially described in an equation is often referred to as the primary hyperbola.
    • For example, x2a2y2b2=1 or y2b2x2a2=1.
  2. Conjugate Hyperbola:

    • The hyperbola obtained by switching the positions of x and y and negating one of the terms is the conjugate hyperbola.
    • If the primary hyperbola is x2a2y2b2=1, the conjugate hyperbola is y2b2x2a2=1.

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 e, the conjugate hyperbola also has eccentricity e.

V. Length of Axes:

  • The length of the major axis of the primary hyperbola is 2a, while the length of the minor axis is 2b.
  • The length of the major axis of the conjugate hyperbola is 2b, and the length of the minor axis is 2a.

VI. Example:

  • Consider the primary hyperbola x24y29=1.
  • The conjugate hyperbola is y29x24=1.
  • Both hyperbolas have the same center but differ in the orientation of their major and minor axes.