Focus of a conic

Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$. Suppose that $Q$ is an ellipse or a hyperbola 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, its centre is its only focus. If $Q$ is a parabola 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, 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 $(0, 1)$.