The eccentricity of a rectangular hyperbola is

The general equation of a hyperbola is \$(\frac{x^2}{a^2}-\frac{y^2}{b^2}=$1), where $a$ represents the transverse of hyperbola and $b$ represents the conjugate of the hyperbola.
The eccentricity of a hyperbola is \$(e=\frac{\sqrt{a^2+b^2}}{a} \cdots($1)).
Since a rectangular hyperbola is defined as a hyperbola whose transverse axis is identical to its conjugate axis, $a=b$.
Substitute $a=b$ into (1).
($$\begin{aligned}<br>& e=\frac{\sqrt{a^2+b^2}}{a}\\<br>& \Rightarrow e=\frac{\sqrt{a^2+a^2}}{a}\\<br>& \Rightarrow e=\frac{\sqrt{2 a^2}}{a}\\<br>& \Rightarrow e=\frac{\sqrt{2} a}{a}\\<br>& \Rightarrow e=\sqrt{2}<br>\end{aligned}$$)