What is the Difference Between Hyperbola and Rectangular Hyperbola?

The main difference between a hyperbola and a rectangular hyperbola lies in the lengths of their axes and the orientation of their asymptotes.

In a hyperbola, the transverse axis and the conjugate axis have different measures. The equation of a hyperbola is given by $$x^2/a^2 - y^2/b^2 = 1$$. A rectangular hyperbola, on the other hand, has both the transverse axis and the conjugate axis of equal lengths. The equation of a rectangular hyperbola is given by $$x^2 - y^2 = a^2$$. This type of hyperbola is also called an equilateral hyperbola or right hyperbola.

Key differences between a hyperbola and a rectangular hyperbola include:

• Asymptotes: In a rectangular hyperbola, the asymptotes are perpendicular to each other, while in a general hyperbola, the asymptotes are not necessarily perpendicular.
• Eccentricity: The eccentricity of a rectangular hyperbola is $$\sqrt{2}$$, while the eccentricity of a general hyperbola can have any value between 1 and $$\infty$$.
• Equation: The equation of a general hyperbola is given by $$x^2/a^2 - y^2/b^2 = 1$$, while the equation of a rectangular hyperbola is given by $$x^2 - y^2 = a^2$$.
• Branches: A hyperbola consists of two distinct branches called connected components, with the closest points on the two branches being called vertices. A rectangular hyperbola also has two branches, but they are related to the asymptotes in a different way.

Comparative Table: Hyperbola vs Rectangular Hyperbola

Here is a table comparing the differences between a hyperbola and a rectangular hyperbola:

A hyperbola is a type of conic section with the general equation (x^2/a^2) - (y^2/b^2) = 1, and it consists of two distinct branches called connected components. The closest points on the two branches are called vertices, and the lines that pass through these two points are called asymptotes.

A rectangular hyperbola is a special case of a hyperbola, where a = b in the equation of the hyperbola, resulting in the equation (x^2 - y^2)/a^2 = 1. In a rectangular hyperbola, the asymptotes are perpendicular to each other, and the eccentricity is equal to √2. The vertex coordinates and focus coordinates of a rectangular hyperbola are also different from those of a general hyperbola.