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  "body": "\n## Definition\n\nLet $F_1$ and $F_2$ be points of the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $a$ be a real number with $2a \u003e d(F_1, F_2)$.\nThe ellipse with the foci $F_1$, $F_2$ and the semi-major axis $a$ is the set\n$$\\{\\, P \\in \\mathbb{R}^2 \\;:\\; d(P, F_1) + d(P, F_2) = 2a \\,\\}.$$\n\n## Remarks\n\nLet $c = \\frac{1}{2} d(F_1, F_2)$ and $b = \\sqrt{a^2 - c^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the ellipse is the set of the points with $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$.\nIf $F_1 = F_2$, the ellipse is the [[circle|060W5CBZP6]] with the centre $F_1$ and the radius $a$.\nThe [[eccentricity|0HVX863SAJ]] of the ellipse is $e = c / a$, with $0 \\leq e \u003c 1$. The ellipse bounds a region of area $\\pi a b$.\n\n## Explanations\n\n### L1 human\n\nFix two pins in a board and tie a loose string to both. Pull the string tight with a pencil and move it round: the pencil draws an ellipse. The length of the string is the constant sum $2a$ of the distances to the two pins.\n",
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    "name": "Ellipse",
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  "msc": [
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  "name": "Ellipse",
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