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  "body": "\n## Definition\n\nLet $F$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$, and let $\\ell$ be a [[line|0MP9BCF6DP]] that does not contain $F$. Let $d(P, F)$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $d(P, \\ell)$ be the [[distance from $P$ to $\\ell$|01AAYNSBN6]].\nThe parabola with the focus $F$ and the directrix $\\ell$ is the set\n$$\\{\\, P \\in \\mathbb{R}^2 \\;:\\; d(P, F) = d(P, \\ell) \\,\\}.$$\n\n## Remarks\n\nFor $F = (a, 0)$ with $a \u003e 0$ and $\\ell$ the line $x = -a$, the parabola is the set of the points with $y^2 = 4 a x$.\nFor real numbers $p \\neq 0$, $q$ and $r$, the graph of $y = p x^2 + q x + r$ is a parabola.\nThe [[eccentricity|0HVX863SAJ]] of each parabola is $1$. Any two parabolas are similar.\nMenaechmus studied conic sections in the 4th century BC. Archimedes found the area of a parabolic segment, and the name is due to Apollonius.\n\n## Explanations\n\n### L1 human\n\nA 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 $y = x^2 / 4$.\u003e\u003e\u003e\n",
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