Parabola

Definition

Let $F$ be a point of the Euclidean plane $\mathbb{R}^2$, and let $\ell$ be a line that does not contain $F$. Let $d(P, F)$ be the Euclidean distance, and let $d(P, \ell)$ be the distance from $P$ to $\ell$. The parabola with the focus $F$ and the directrix $\ell$ is the set $$\{\, P \in \mathbb{R}^2 \;:\; d(P, F) = d(P, \ell) \,\}.$$

Remarks

For $F = (a, 0)$ with $a > 0$ and $\ell$ the line $x = -a$, the parabola is the set of the points with $y^2 = 4 a x$. For real numbers $p \neq 0$, $q$ and $r$, the graph of $y = p x^2 + q x + r$ is a parabola. The eccentricity of each parabola is $1$. Any two parabolas are similar. Menaechmus studied conic sections in the 4th century BC. Archimedes found the area of a parabolic segment, and the name is due to Apollonius.

Explanations

L1 human

A parabola is the set of points that are as far from a point as from a line. For the focus $(0, 1)$ and the line $y = -1$, the parabola is $y = x^2 / 4$.>>>