What is the eccentricity of rectangular hyperbola?
A. $\sqrt 2 $
B. \[\sqrt 3 \]
C. $\sqrt 5 $
D. $\sqrt 6 $

Hint: Rectangular hyperbola also called right hyperbola in which the transverse axis and conjugate axes are equal. The axes of the rectangular hyperbola are perpendicular to each other.

Formula used: Equation of hyperbola: $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$
Eccentricity of hyperbola: \[{a^2}({e^2} - 1) = {b^2}\]

Complete step by step solution: General equation of hyperbola: $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$
As \[2a = 2b\] therefore, \[a = b\]
 $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{a^2}}} = 1$
${x^2} - {y^2} = 1$
${x^2} = {y^2}$
$x = \pm y$
Eccentricity is given by formula
\[{a^2}({e^2} - 1) = {b^2}\]
${(ae)^2} = {a^2} + {b^2}$
${(ae)^2} = {a^2} + {a^2}$
${(ae)^2} = 2{a^2}$
\[{a^2}{e^2} = 2{a^2}\]
\[e = \sqrt 2 \]

Thus, Option (A) is correct.

Note: On rotating hyperbola by an angle of $- 45^{\circ}$ about the same origin, then the equation of the rectangular hyperbola \[{x^2}\;-{\text{ }}{y^2}\; = {\text{ }}{a^{2\;}}\] is reduced to \[xy{\text{ }} = {\text{ }}\dfrac{{{a^2}}}{2}\] or \[xy{\text{ }} = {\text{ }}{c^2}\]. The vertices of the hyperbola are given by \[\left( {c,{\text{ }}c} \right)\] and \[\left( { - c,{\text{ }} - c} \right)\] and the focus is \[\left( {\surd 2c,\;\surd 2c} \right)\] and \[\left( { - \surd 2c,{\text{ }} - \surd 2c} \right)\].The axis are the coordinate axis when \[xy{\text{ }} = {\text{ }}{c^2}\].