# Hyperbola vs. Rectangular Hyperbola: What's the Difference?

Edited by Aimie Carlson || By Harlon Moss || Published on May 3, 2024
A hyperbola is a type of conic section with two branches, while a rectangular hyperbola is a specific hyperbola with equal asymptote angles, forming a rectangle.

## Key Differences

A hyperbola is an open curve formed by intersecting a cone with a plane at an angle steeper than the cone's side, resulting in two separate, mirror-image branches. Its defining feature is its two asymptotes, lines that the curve approaches but never touches. A rectangular hyperbola, a specific type of hyperbola, is characterized by having asymptotes that are perpendicular to each other, thus forming a right angle.
In a hyperbola, the asymptotes provide a skeletal outline of the curve, guiding its shape but never intersecting with the branches. The hyperbola can have a wide range of orientations and shapes, determined by the relative lengths of its semi-major and semi-minor axes, and the angle of the asymptotes. In contrast, the rectangular hyperbola has a more specific structure due to its perpendicular asymptotes. This constraint fixes the relationship between the axes of the hyperbola, making the semi-major and semi-minor axes equal in length. This equality results in the branches of the rectangular hyperbola having a consistent shape, regardless of the size of the hyperbola.
All rectangular hyperbolas are hyperbolas, not all hyperbolas are rectangular hyperbolas. The distinction lies in the orientation and relationship of the asymptotes: a rectangular hyperbola always has perpendicular asymptotes, whereas a general hyperbola does not have this restriction. This difference fundamentally affects their geometric properties and equations representing them.

## Comparison Chart

### Asymptotes

Can be at any angle to each other
Are perpendicular to each other

### Eccentricity

Varies, always greater than 1
Exactly √2

### Axes Relationship

Semi-major and semi-minor axes differ
Semi-major and semi-minor axes are equal

### Shape

Varies based on axes and asymptote angle
Consistent shape due to equal axes

## Hyperbola and Rectangular Hyperbola Definitions

#### Hyperbola

Has asymptotes that guide its shape.
The hyperbola never touches its asymptotes, no matter how far it extends.

#### Rectangular Hyperbola

Its axes are of equal length.
In a rectangular hyperbola, the semi-major and semi-minor axes coincide.

#### Hyperbola

Defined by its eccentricity, greater than one.
The hyperbola's eccentricity dictates its openness.

#### Rectangular Hyperbola

Symmetrical about both the x and y axes.
The symmetry of a rectangular hyperbola simplifies its analysis.

#### Hyperbola

A conic section with two separate, unbounded branches.
The path of a comet around the sun can be a hyperbola.

#### Rectangular Hyperbola

Often used to represent inverse relationships.
Boyle's law in physics is depicted using a rectangular hyperbola.

#### Hyperbola

Represents the difference of distances to two foci is constant.
Each point on the hyperbola maintains a constant difference from the two foci.

#### Rectangular Hyperbola

Has an eccentricity of √2.
The unique eccentricity of √2 defines the rectangular hyperbola.

#### Hyperbola

Can be expressed in terms of its semi-major and semi-minor axes.
The length of the hyperbola's axes determines its spread.

#### Hyperbola

A plane curve having two branches, formed by the intersection of a plane with both halves of a right circular cone at an angle parallel to the axis of the cone. It is the locus of points for which the difference of the distances from two given points is a constant.

#### Hyperbola

(geometry) A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. The function y(x) = 1/x draws a hyperbola. Category:en:Curves Category:en:Functions

#### Hyperbola

A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.

#### Hyperbola

An open curve formed by a plane that cuts the base of a right circular cone

## FAQs

#### Can a hyperbola have any eccentricity greater than one?

Yes, a hyperbola's eccentricity is always greater than one, but the exact value varies.

#### What is a hyperbola?

A hyperbola is a type of conic section with two separate, unbounded branches that are mirror images of each other.

#### What is the eccentricity of a rectangular hyperbola?

The eccentricity of a rectangular hyperbola is exactly √2.

#### How do the asymptotes of a hyperbola and a rectangular hyperbola differ?

A hyperbola's asymptotes can be at any angle, while a rectangular hyperbola's asymptotes are always perpendicular.

#### Why are the axes of a rectangular hyperbola equal in length?

The perpendicular nature of the asymptotes in a rectangular hyperbola results in equal semi-major and semi-minor axes.

#### What defines a rectangular hyperbola?

A rectangular hyperbola is defined by its perpendicular asymptotes, resulting in equal semi-major and semi-minor axes.

#### Is every hyperbola symmetrical?

Yes, every hyperbola is symmetrical about its principal axis and the line perpendicular to it through the center.

#### What is an example of a rectangular hyperbola in real life?

Boyle's law in physics, which describes the inverse relationship between pressure and volume of a gas, can be represented by a rectangular hyperbola.

#### How does the eccentricity of a hyperbola affect its openness?

The greater the eccentricity of a hyperbola, the more open and less curved its branches are.

#### What role do the foci play in a hyperbola?

The foci of a hyperbola are two fixed points used in its definition; the difference in distances from any point on the hyperbola to the foci is constant.

#### How is the shape of a hyperbola determined?

The shape of a hyperbola is determined by its semi-major and semi-minor axes and the angle of its asymptotes.

#### How is a rectangular hyperbola used in economics?

In economics, a rectangular hyperbola can represent constant product curves, such as in the Cobb-Douglas production function.

#### What are the asymptotes of a hyperbola?

The asymptotes of a hyperbola are straight lines that the branches of the hyperbola approach but never intersect.

#### Can a rectangular hyperbola be rotated?

A rectangular hyperbola can be rotated, but its defining characteristic of perpendicular asymptotes remains unchanged.

#### What is the significance of the axes in a hyperbola?

The axes of a hyperbola define its orientation and the extent of its branches.

#### What distinguishes a hyperbola from other conic sections?

A hyperbola is distinguished by its open, unbounded branches and its definition involving the difference of distances to two foci.

#### How does a rectangular hyperbola relate to inverse functions?

A rectangular hyperbola often represents inverse relationships, where the product of two variables is constant.

#### Why is a rectangular hyperbola considered 'rectangular'?

It's termed 'rectangular' because the asymptotes intersect at right angles, forming a rectangle if extended.

#### Can the properties of a hyperbola be applied to real-world problems?

Yes, the properties of hyperbolas are applied in various fields, including astronomy, physics, and engineering, to solve practical problems.