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  "body": "\n## Statement\n\nLet $Q = \\{\\, (x, y) \\in \\mathbb{R}^2 : A x^2 + B x y + C y^2 + D x + E y + F = 0 \\,\\}$ be a non-degenerate [[conic section|00HJCMC2CD]]. Let $\\Delta = B^2 - 4AC$ be the discriminant.\nThen $Q$ is\n- an [[ellipse|0VKCCEXBJ1]] if and only if $\\Delta \u003c 0$,\n- a [[parabola|04S2GMHX82]] if and only if $\\Delta = 0$,\n- a [[hyperbola|0WN5G117YC]] if and only if $\\Delta \u003e 0$.\n\nMoreover, $Q$ is a [[circle|060W5CBZP6]] if and only if $A = C$ and $B = 0$.\n\n## Remarks\n\nThe discriminant is $-4$ times the determinant of $\\begin{pmatrix} A \u0026 B/2 \\\\ B/2 \u0026 C \\end{pmatrix}$, the matrix of the quadratic part.\nA rotation or a translation of the coordinates does not change $\\Delta$ or $A + C$. So the type of a conic does not depend on the choice of Cartesian coordinates.\nA hyperbola with $A + C = 0$ is rectangular: its asymptotes are perpendicular.\n\n## Explanations\n\n### L1 human\n\nThe terms of degree $2$ decide the shape of a conic. For $x^2 + y^2 = 1$, $\\Delta = -4 \u003c 0$ and we get a circle, a special ellipse. For $y = x^2$, $\\Delta = 0$ and we get a parabola. For $x y = 1$, $\\Delta = 1 \u003e 0$ and we get a hyperbola.\n",
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