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  "body": "\n## Definition\n\nLet $A, B, C, D, E, F$ be real numbers, with $A$, $B$ and $C$ not all zero. Let\n$$M = \\begin{pmatrix} A \u0026 B/2 \u0026 D/2 \\\\ B/2 \u0026 C \u0026 E/2 \\\\ D/2 \u0026 E/2 \u0026 F \\end{pmatrix}.$$\nThe set $Q = \\{\\, (x, y) \\in \\mathbb{R}^2 \\;:\\; A x^2 + B x y + C y^2 + D x + E y + F = 0 \\,\\}$ in [[Cartesian coordinates|0RWKYX8339]] is a non-degenerate conic section if $\\det M \\neq 0$ and $Q$ is not empty.\n\n## Remarks\n\nIf $\\det M = 0$, the set $Q$ is a degenerate conic: two lines, one line, one point or the empty set.\nEach non-degenerate conic is an [[ellipse|0VKCCEXBJ1]], a [[parabola|04S2GMHX82]] or a [[hyperbola|0WN5G117YC]]. The [[discriminant $B^2 - 4AC$|0N5NR32NQG]] tells which one.\nEquivalently, a non-degenerate conic is the intersection of a double cone with a plane that does not pass through the apex.\nApollonius of Perga wrote a systematic work on conic sections, about 200 BC.\n\n## Explanations\n\n### L1 human\n\nCut a double ice cream cone with a flat knife. Depending on the tilt of the knife, the cut edge is a closed oval, a U shape or a curve with two branches. These are the ellipse, the parabola and the hyperbola, and they all have equations of degree $2$.\n",
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        "url": "https://en.wikipedia.org/wiki/Conic_section"
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