# 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, $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ or $\frac{{y}^{2}}{{b}^{2}}-\frac{{x}^{2}}{{a}^{2}}=1$.
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 $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$, the conjugate hyperbola is $\frac{{y}^{2}}{{b}^{2}}-\frac{{x}^{2}}{{a}^{2}}=1$.

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 $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}=1$.
• The conjugate hyperbola is $\frac{{y}^{2}}{9}-\frac{{x}^{2}}{4}=1$.
• Both hyperbolas have the same center but differ in the orientation of their major and minor axes.