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  "body": "\n## Definition\n\nLet $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\\mathbb{R}^2$ that is not a circle. Let $F$ be a [[focus|0RBKKNH4GV]] of $Q$. For a point $P$ and a [[line|0MP9BCF6DP]] $\\ell$, let $d(P, \\ell)$ be the [[distance from $P$ to $\\ell$|01AAYNSBN6]].\nA directrix of $Q$ for the focus $F$ is a line $\\ell$ that does not contain $F$, such that for a real number $e \u003e 0$:\n$$Q = \\{\\, P \\in \\mathbb{R}^2 \\;:\\; d(P, F) = e \\, d(P, \\ell) \\,\\}.$$\n\n## Remarks\n\nEach focus of $Q$ has exactly one directrix, and the number $e$ is the [[eccentricity|0HVX863SAJ]] of $Q$. The conic is an ellipse if $e \u003c 1$, a parabola if $e = 1$ and a hyperbola if $e \u003e 1$.\nFor the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ or the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, the directrices are the lines $x = \\pm a^2 / c$, for the foci $(\\pm c, 0)$.\nA circle has no directrix in the Euclidean plane.\n\n## Explanations\n\n### L1 human\n\nFor a parabola, the directrix is the line in its definition: each point is as far from the focus as from the directrix. For an ellipse, each point is closer to the focus than to the directrix, by a fixed factor $e \u003c 1$. For a hyperbola, the factor is greater than $1$.\n",
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        "locator": "Eccentricity, focus and directrix",
        "url": "https://en.wikipedia.org/wiki/Conic_section"
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        "locator": "By the directrix property",
        "url": "https://en.wikipedia.org/wiki/Hyperbola"
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