Exploring the Graph of x2y2 1 and Its Variants

The equation x2y2 1 is a fascinating example of a curve-pair or hyperbola-pair. This article delves into its properties and graph, as well as related hyperbolas that share similar characteristics.

Equation and Graph of x2y2 1

The equation x2y2 1 can be rewritten as:

xy2 1

xy2 - 1 0

xy ±1

y ±1/x

These transformations reveal that the equation describes a pair of hyperbolas. The hyperbolas are:

xy 1 xy -1

The solution xy ± 1 indicates that the graph consists of two branches that are reciprocals of each other, centered at the origin. The standard form of these hyperbolas is:

x y 1 [/itex]

and

x y - 1 [/itex]

Graphically, these hyperbolas approach the coordinate axes asymptotically, with the asymptotes forming a right angle.

The Hyperbola of Fermat

The specific case of y2 x-2 is a type of hyperbola known as the Hyperbola of Fermat. More generally, this can be written as:

y m x n [/itex]

This equation represents a broader class of hyperbolas, where the exponents m and n can be positive or negative. In the case of the Hyperbola of Fermat, the negative exponent indicates a reciprocal relationship between x and y2.

Related Equations and Graphs

Related equations such as x2y2 1, x3y2 1, and x2y 1 also represent curves that share similar properties with the hyperbolas discussed above. These graphs can be understood as variants of the fundamental hyperbola xy 1, where the first and third include an additional factor:

xy2 1

x2y 1

These forms introduce a modification in the slopes and the behavior of the asymptotes, but the basic structure remains a pair of hyperbolas.

Conclusion

In summary, the equation x2y2 1 describes a fundamental pair of hyperbolas with reciprocal properties. The study of such equations provides insight into the broader class of hyperbolas and their geometric properties. Understanding these relationships is crucial for advanced mathematical analysis and applications in various scientific fields.