# 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