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**

**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**

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