---
tag: 0P234N3M28
kind: definition
name: Hyperbola
msc: [51N20]
defines: 0WN5G117YC
uses: [0XW7V9ZZ05, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbola
    locator: As locus of points
  - url: https://en.wikipedia.org/wiki/Hyperbola
    locator: Asymptotes
---

## Definition

Let $F_1$ and $F_2$ be distinct points of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. 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|0HVX863SAJ]] 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 $2a$.