Hyperbola

Definition

Let $F_1$ and $F_2$ be distinct points of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the Euclidean distance. Let $a$ be a real number with $0 < 2a < d(F_1, F_2)$. The hyperbola with the foci $F_1$, $F_2$ and the semi-major axis $a$ is the set $$\{\, P \in \mathbb{R}^2 \;:\; |d(P, F_1) - d(P, F_2)| = 2a \,\}.$$

Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{c^2 - a^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the hyperbola is the set of the points with $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$. This hyperbola has the two asymptotes $y = \pm \frac{b}{a} x$. Its eccentricity is $e = c / a > 1$. For $a = b$, the asymptotes are perpendicular, and the hyperbola is rectangular. The graph of $y = 1/x$ is a rectangular hyperbola.

Explanations

L1 human

A hyperbola has two separate branches, like two opposite bows. On one branch, the points are nearer to one focus; on the other branch, they are nearer to the other focus. The difference of the two distances is always $2a$.