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,

Rectangular Hyperbola

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?

Equilateral Hyperbola

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

• Definition of Hyperbola
• Standard Equation of an Hyperbola
  
• Vertex of the Hyperbola
• Centre of the Hyperbola
• Transverse and Conjugate Axis of the Hyperbola
• Two Foci and Two Directrices of the Hyperbola
• Latus Rectum of the Hyperbola
• Position of a Point with Respect to the Hyperbola
• Conjugate Hyperbola
• Rectangular Hyperbola
• Parametric Equation of the Hyperbola
• Hyperbola Formulae
• Problems on Hyperbola

11 and 12 Grade Math

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