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  "body": "\n## Definition\n\nLet $F_1$ and $F_2$ be distinct points of the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $a$ be a real number with $0 \u003c 2a \u003c d(F_1, F_2)$.\nThe hyperbola 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 $b = \\sqrt{c^2 - a^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the hyperbola is the set of the points with $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1$.\nThis hyperbola has the two asymptotes $y = \\pm \\frac{b}{a} x$. Its [[eccentricity|0HVX863SAJ]] is $e = c / a \u003e 1$.\nFor $a = b$, the asymptotes are perpendicular, and the hyperbola is rectangular. The graph of $y = 1/x$ is a rectangular hyperbola.\n\n## Explanations\n\n### L1 human\n\nA hyperbola has two separate branches, like two opposite bows. On one branch, the points are nearer to one focus; on the other branch, they are nearer to the other focus. The difference of the two distances is always $2a$.\n",
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  "name": "Hyperbola",
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