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 xaxis and its length = 2a.
The conjugate axis of the hyperbola (i) is along yaxis 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 coordinates of foci and the length of semilatus 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 coordinates of its foci are (± ae, 0) = (± 5√2, 0).
The length of semilatus 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
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