Rectangular Hyperbola is a hyperbola in which the transverse and conjugate axes are equal. i.e. in the case of rectangular hyperbola a = b = 1. The asymptote of the rectangular hyperbola is y = ±x. Also, the asymptotes of a rectangular hyperbola are perpendicular.
In this article, we will explore the rectangular hyperbola in depth along with its standard equation, eccentricity, asymptotes, and parametric equation.
What is a Rectangular Hyperbola?
The rectangular hyperbola is the hyperbola in which the lengths of the transverse axis and conjugate axis are the same. The eccentricity of the rectangular hyperbola is √2. The length of the transverse axis 2a and the length of the conjugate axis 2b are equal. The foci of the rectangular hyperbola is (± a√2, 0). The image below shows the rectangular hyperbola.
Rectangular Hyperbola Shape
The shape of a rectangular hyperbola, when described in more geometric terms, is defined by two distinct branches that extend indefinitely, bending away from each other in opposing quadrants. Each branch is a mirror image of the other across the origin when the hyperbola is centered at the origin.
Rectangular Hyperbola Equation
General equation of the Rectangular Hyperbola centered at origin (0, 0) is,
x2 - y2 = a2
If center shifts to (x0, y0), then equation of rectangular hyperbola becomes
(x - x0)2 - (y - y0)2 = a2
Parametric Equation of Rectangular Hyperbola
The parametric equation of the rectangular hyperbola is,
x = a secθ
y = a tanθ
Rectangular Hyperbola Graph
A rectangular hyperbola is a type of hyperbola that is specifically defined as having the property that the asymptotes are perpendicular to each other, forming a right angle. Graph of a Rectangular Hyperbola with equation xy = c2 where c is a constant that determines the scale of the hyperbola.
Some of the rectangular hyperbola formulas and equations are listed below
- Eccentricity of the rectangular hyperbola is √2.
- Asymptotes equation of the rectangular hyperbola is y = ±x or x2 - y2 = 0.
- Standard equation of the rectangular hyperbola is x2 - y2 = a2
- Parametric equation of the rectangular hyperbola is x = asecθ, y = atanθ
Eccentricity of Rectangular Hyperbola
The equation of rectangular hyperbola is, x2 - y2 = a2. Now we know that eccentricity of the hyperbola is,
e = √(1 + b2/a2)
In case of rectangular hyperbola, a = b = 1. Now,
e = √(1 + b2/a2) = √(1 + 1/1) =√(2)
Asymptotes of a Rectangular Hyperbola
Asymptote are the lines that connects the curve at infinity. In case of rectangular hyperbola the equation of asymptote is,
Asymptotes of a Rectangular Hyperbola are Perpendicular.
Properties of Rectangular Hyperbola
Rectangular Hyperbola has various properties and some of the important properties of the rectangular hyperbola are,
- Lengths of the transverse and conjugate axis are equal in the rectangular hyperbola.
- Asymptotes of the rectangular hyperbola are perpendicular to each other.
- Conjugate of the rectangular hyperbola is also a rectangular hyperbola.
- Hyperbola whose asymptotes are perpendicular are called as the right hyperbola or equilateral hyperbola.
Shifting of origin of the rectangular hyperbola is a very important concept for students, suppose we take a rectangular hyperbola and the coordinate of any point is A(x, y) and its origin is rotated anticlockwise by π/4 then the in new coordinate system the point A is transformed to B(X, Y) where,
- X = x.cosα - y.sinα = x.cos(π/4) + y.sin(π/4) = (x - y)/√(2)...(i)
- Y = x.sinα + y.cosα = x.sin(π/4) + y.cos(π/4) = (x + y)/√(2)...(ii)
Now equation of the rectangular hyperbola is,
X2 - Y2 = a2
⇒ {(x - y)/√(2)}2 - {(x + y)/√(2)}2 = a2
⇒ (x2 + y2 - 2xy)/2 - (x2 + y2 + 2xy)/2 = a2
⇒ -4xy/2 = a2
⇒ xy = a2/-2
Let, c = -1/2a2 then equation becomes
xy = constant
Now the various formulas for the rectangular hyperbola xy = c2 with parameter 't' and any point (ct, c/t) are,
Foci
| (±√(2)c, ±√(2)c)
|
Eccentricity
| √(2)c
|
Transverse Axis
| 2√(2)c
|
Directrices
| x + y = ±√(2)c
|
Asymptotes
| |
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Examples on Rectangular Hyperbola
Example 1: Given that the length of the transverse axis is 18 units and coordinate axes as its axis then find the equation of the rectangular hyperbola.
Solution:
Length of Transverse Axis = 2a = 18 units
a = 9 units
Equation of Rectangular Hyperbola is given, x2 - y2 = a2
x2 - y2 = 92
Equation of the given Rectangular Hyperbola is, x2 - y2 = 81
Example 2: Find the foci of the rectangular hyperbola whose equation is x2 - y2 = 25.
Solution:
Equation of Rectangular Hyperbola is, x2 - y2 = a2...(i)
Given Equation,
x2 - y2 = 52
Comparing Equation (i) and (ii)
a = 5
Foci of Rectangular Hyperbola is (± a√2, 0)
So, Foci of Given Rectangular Hyperbola is (± 5√2, 0)
Example 3: Find the length of transverse axis of the rectangular hyperbola whose equation is x2 - y2 = 9.
Solution:
Equation of Rectangular Hyperbola is, x2 - y2 = a2...(i)
Given Equation,
x2 - y2 = 32
Comparing eq. (i) and (ii)
a = 3
Length of Transverse Axis of Rectangular Hyperbola = 2a
So, length of transverse axis of given rectangular hyperbola = 2(3) = 6 units.
Example 4: Find the length of latus rectum of the rectangular hyperbola whose equation is x2 - y2 = 36.
Solution:
Equation of Rectangular Hyperbola is, x2 - y2 = a2...(i)
Given Equation,
x2 - y2 = 62
Comparing eq. (i) and (ii)
a = 6
Length of Latus Rectum of Rectangular Hyperbola = 2a
So, Length of Latus Rectum of the given rectangular hyperbola = 2(6) = 12 units
Practice Questions on Rectangular Hyperbola
Q1. Given that the length of the transverse axis is 6 units and coordinate axes as its axis then find the equation of the rectangular hyperbola.
Q2. Find the foci of the rectangular hyperbola whose equation is x2 - y2 = 121.
Q3. Find the length of transverse axis of the rectangular hyperbola whose equation is x2 - y2 = 100.
Q4. Find the length of latus rectum of the rectangular hyperbola whose equation is x2 - y2 = 196.
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