A glimpse of Hyperbola

Let's take the rectangular hyperbola $x^2 - y^2 = k^2$ as an example. The eccentricity of any hyperbola is given by $e = \frac{\sqrt{a^{2} + b^{2}}}{a}$ , and our rectangular hyperbola (which is one whose asymptotes are perpendicular to each other) has $a = b = k$ . The eccentricity is computed as $e = \frac{\sqrt{k^{2} + k^{2}}}{k} = \frac{k \cdot \sqrt{2}}{k} = \boxed{\sqrt{2}}$ for all $k \in \mathbb{R}.$