---
tag: 0RV3AEXPT6
kind: definition
name: Focus of a conic
msc: [51N20]
defines: 0RBKKNH4GV
uses: [0VKCCEXBJ1, 0WN5G117YC, 04S2GMHX82, 060W5CBZP6]
sources:
  - url: https://en.wikipedia.org/wiki/Focus_(geometry)
    locator: Defining conics in terms of two foci
  - url: https://en.wikipedia.org/wiki/Focus_(geometry)
    locator: Defining conics in terms of a focus and a directrix
---

## Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$.
Suppose that $Q$ is an [[ellipse|0VKCCEXBJ1]] or a [[hyperbola|0WN5G117YC]] with the foci $F_1$ and $F_2$, as in their definitions. Then $F_1$ and $F_2$ are the foci of $Q$. If $Q$ is a [[circle|060W5CBZP6]], its centre is its only focus.
If $Q$ is a [[parabola|04S2GMHX82]] with the focus $F$ and a directrix, as in its definition, then $F$ is the only focus of $Q$.

## Remarks

The foci are determined by the conic. For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, the foci are $(\pm c, 0)$ with $c^2 = a^2 - b^2$.
For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the foci are $(\pm c, 0)$ with $c^2 = a^2 + b^2$.
Let $Q$ be a conic that is not a circle, and let $F$ be a focus of $Q$. Then $Q$ is the set of the points whose distances to $F$ and to a line, the [[directrix|0E5CXYNEPX]], have a constant ratio $e > 0$.

## Explanations

### L1 human

The foci are the two pins in the string construction of an ellipse. A parabolic mirror sends the light of a lamp at its focus out in parallel rays. For the parabola $y = x^2/4$, the focus is $(0, 1)$.