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:

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 $\frac{{y}^{2}}{{b}^{2}}\frac{{x}^{2}}{{a}^{2}}=1$.

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.