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Rectangular Hyperbola

Rectangular Hyperbola is a hyperbola having the transverse axis and the conjugate of 2a units and conjugate axis of 2b units of equal length. The eccentricity of a rectangular hyperbola is √2, and the equation of a rectangular hyperbola is x² - y² = a².

Let us learn more about the equation, eccentricity, asymptotes, parametric equation of a rectangular hyperbola.

1.	What Is A Rectangular Hyperbola?
2.	Properties of Rectangular Hyperbola
3.	Examples on Rectangular Hyperbola
4.	Practice Questions
5.	FAQs on Rectangular Hyperbola

What Is A Rectangular Hyperbola?

A rectangular hyperbola is a hyperbola having the transverse axis and the conjugate axis of equal length. The arcs of a rectangular hyperbola is the same as the arc of a circle. For a rectangular hyperbola having the transverse axis of length 2a and the conjugate axis of length 2b, we have 2a = 2b, or a = b. The general equation of a rectangular hyperbola is x² - y² = a^2..

The equation of asymptotes of a rectangular hyperbola is y = + x or x² - y² = 0.The axes or the asymptotes of the rectangular hyperbola are perpendicular to each other. The rectangular hyperbola is related to a hyperbola in a similar form as the circle is related to an ellipse. The eccentricity of a rectangular hyperbola is √2. The graph of the equation y = 1/x is similar to the graph of a rectangular hyperbola.

Properties of Rectangular Hyperbola

The rectangular hyperbola is similar to a regular hyperbola, and the only difference is the different lengths of the transverse axis and conjugate axis in a hyperbola, and these lengths are equal in a rectangular hyperbola The following are some of the important properties of a rectangular hyperbola.

The eccentricity of a rectangular hyperbola is equal to √2.
The transverse axis and the conjugate axis in a rectangular hyperbola is of equal length.
The asymptotoes of a rectangular hyperbola is y = + x or x² - y² = 0.
The asymptotes of a rectangular hyperbola are perpendicular to each other.
The conjugate of a rectangular hyperbola x² - y² = a² is also a rectangular hyperbola x² - y² = -a².
The parametric form of representation of a rectangular hyperbola has the coordinates x = aSecθ, y = aTanθ.

Related Topics

The following topics help in a better understanding of rectangular hyperbola.

Examples on Rectangular Hyperbola

Example 1: Find the equation of a rectangular hyperbola having the transverse axis of 10 units, and with the coordinate axes as its axis.

Solution:

Here it is given that the coordinate axes is the axes of the hyperbola. Hence the required equation of the rectangular hyperbola is x² - y² = a².

The length of the transverse axis = 2a = 10 units or we have a = 5.

Hence the equation of the rectangular hyperbola is x² - y²= 5^2, or x² - .y² = 25

Therefore the required equation of the rectangular hyperbola is x² - y² = 25
Example 2: Find the foci, length of the transverse axis, length of the latus rectum of the rectangular hyperbola x² - y² = 16.

Solution:

The given equation of the rectangular hyperbola is x² - y² = 16

This on comparing with the standard equation of the rectangular hyperbola x² - y² = a², we have a² = 16 or a = 4.

The eccentricity of the rectangular hyperbola is e = √2

Foci = (ar, o) = (+4√2, 0).

Length of transverse axes = 2a = 2(4) = 8.

Length of the latus rectum = 2a = 2(4) = 8.

Therefore, the foci of the rectangular hyperbola is (+4√2, 0), and the length of the transverse axis, and the length of the latus rectum is 8 units.

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Practice Questions on Rectangular Hyperbola

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FAQs on Rectangular Hyperbola

What Is A Rectangular Hyperbola?

A rectangular hyperbola is a hyperbola having the transverse axis and the conjugate axis of equal length. For a rectangular hyperbola we have 2a = 2b, or a = b. The general equation of a rectangular hyperbola is x² - y² = a².

What Is the Equation of a Rectangular Hyperbola?

The equation of a rectangular hyperbola is x² - y² = a², where 'a' is the length of the semi-major axis of the hyperbola. Here in a rectangular hyperbola both the transverse axes and the conjugate axes are of equal length.

What Is the Formula of a Rectangular Hyperbola?

The formula and the equation of a rectangular hyperbola is the same and is x² - y² = a². Here 'a' is the length of the semi-major axis.

What Is the Difference Between A Hyperbola And A Rectangular Hyperbola?

The difference between the hyperbola and a rectangular hyperbola is the difference in the lengths of the transverse axis and the conjugate axis. In a hyperbola, the transverse axis and the conjugate axis are of different measures, and in a rectangular hyperbola both the transverse axis and the conjugate axes are of equal lengths. The equation of a hyperbola is x²/a² - y²/b² = 1, and the equation of a rectangular hyperbola is x² - y² = a².

Why Is A Hyperbola Called A Rectangular Hyperbola?

The hyperbola is called a rectangular hyperbola because the length of its transverse axis is equal to the length of its conjugate axis, 2a = 2b. The equation of a rectangular hyperbola is x² - y² = a².

What Are the Properties Of A Rectangular Hyperbola?

The following are some of the important properties of a rectangular hyperbola.

The eccentricity of a rectangular hyperbola is equal to √2.
The transverse axis and the conjugate axis in a rectangular hyperbola is of equal length.
The asymptotoes of a rectangular hyperbola is y = + x or x² - y² = 0.
The asymptotes of a rectangular hyperbola are perpendicular to each other

What Is the Parametric Form Of A Rectangular Hyperbola?

The parametric form of representation of a rectangular hyperbola has the coordinates x = aSecθ, y = aTanθ.

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