Rectangular Hyperbola

What is rectangular hyperbola?

When the transverse axis of a hyperbola is equal to its conjugate axis then the hyperbola is called a rectangular or equilateral hyperbola.

The standard equation of the hyperbola \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 ………… (i)

The transverse axis of the hyperbola (i) is along x-axis and its length = 2a.

The conjugate axis of the hyperbola (i) is along y-axis and its length = 2b.

According to the definition of rectangular hyperbola we get, a = b

Therefore, substitute a = b in the standard equation of the hyperbola (i) we get,

\(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 

⇒ \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{a^{2}}\) = 1  

⇒ x\(^{2}\) - y\(^{2}\) = a\(^{2}\), which is the equation of the rectangular hyperbola.

1. Show that the eccentricity of any rectangular hyperbola is √2

Solution:

The eccentricity of the standard equation of the hyperbola \(\frac{x^{2}}{a^{2}}\) - \(\frac{y^{2}}{b^{2}}\) = 1 is b\(^{2}\) = a\(^{2}\)(e\(^{2}\) - 1).

Again, according to the definition of rectangular hyperbola we get, a = b

Therefore, substitute a = b in the eccentricity of the standard equation of the hyperbola (i) we get,

a\(^{2}\) = a\(^{2}\)(e\(^{2}\) - 1)           

⇒ e\(^{2}\) - 1 = 1      

⇒ e\(^{2}\) = 2      

⇒ e = √2   

Thus, the eccentricity of a rectangular hyperbola is √2.


2. Find the eccentricity, the co-ordinates of foci and the length of semi-latus rectum of the rectangular hyperbola x\(^{2}\) - y\(^{2}\) - 25 = 0.

Solution:

Given rectangular hyperbola x\(^{2}\) - y\(^{2}\) - 25 = 0

From the rectangular hyperbola x\(^{2}\) - y\(^{2}\) - 25 = 0 we get,

x\(^{2}\) - y\(^{2}\) = 25

⇒ x\(^{2}\) - y\(^{2}\) = 5\(^{2}\)

⇒ \(\frac{x^{2}}{5^{2}}\) - \(\frac{y^{2}}{5^{2}}\) = 1 

The eccentricity of the hyperbola is

e = \(\sqrt{1 + \frac{b^{2}}{a^{2}}}\)

= \(\sqrt{1 + \frac{5^{2}}{5^{2}}}\), [Since, a = 5 and b = 5]

= √2

The co-ordinates of its foci are (± ae, 0) = (± 5√2, 0).

The length of semi-latus rectum = \(\frac{b^{2}}{a}\) = \(\frac{5^{2}}{5}\) = 25/5 = 5.


3. What type of conic is represented by the equation x\(^{2}\) - y\(^{2}\) = 9? What is its eccentricity?

Solution:

The given equation of the conic x\(^{2}\) - y\(^{2}\) = 9

⇒ x\(^{2}\) - y\(^{2}\) = 3\(^{2}\), which is the equation of the rectangular hyperbola.

A hyperbola whose transverse axis is equal to its conjugate axis is called a rectangular or equilateral hyperbola.

The eccentricity of a rectangular hyperbola is √2.

The Hyperbola






11 and 12 Grade Math

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