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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 x^{2}  y^{2} = a^{2}.
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 x^{2}  y^{2} = a^{2.}.
The equation of asymptotes of a rectangular hyperbola is y = + x or x^{2}  y^{2} = 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 x^{2}  y^{2} = 0.
 The asymptotes of a rectangular hyperbola are perpendicular to each other.
 The conjugate of a rectangular hyperbola x^{2}  y^{2} = a^{2} is also a rectangular hyperbola x^{2}  y^{2} = a^{2}.
 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 x^{2}  y^{2} = a^{2}.
The length of the transverse axis = 2a = 10 units or we have a = 5.
Hence the equation of the rectangular hyperbola is x^{2}  y^{2 }= 5^{2, } or x^{2}  .y^{2} = 25
Therefore the required equation of the rectangular hyperbola is x^{2}  y^{2} = 25

Example 2: Find the foci, length of the transverse axis, length of the latus rectum of the rectangular hyperbola x^{2}  y^{2} = 16.
Solution:
The given equation of the rectangular hyperbola is x^{2}  y^{2} = 16
This on comparing with the standard equation of the rectangular hyperbola x^{2}  y^{2} = a^{2}, we have a^{2} = 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.
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 x^{2}  y^{2} = a^{2}.
What Is the Equation of a Rectangular Hyperbola?
The equation of a rectangular hyperbola is x^{2}  y^{2} = a^{2}, where 'a' is the length of the semimajor 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 x^{2}  y^{2} = a^{2}. Here 'a' is the length of the semimajor 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 x^{2}/a^{2}  y^{2}/b^{2} = 1, and the equation of a rectangular hyperbola is x^{2}  y^{2} = a^{2}.
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 x^{2}  y^{2} = a^{2}.
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 x^{2}  y^{2} = 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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