In implicit form, the equation is: A x^2 + B x y + C y^2 + D x + E y + F = 0 (although there are varients, e.g. writing 2B instead of B or putting F on the right). For some choices of the coefficients A, B, C, D, E, and F the form is degenerate, in which case the equation may have no solution (e.g. A, B, C, D, E are all zero, F is not) or they may represent one or two lines.
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Circles, parabolas, ellipses, and hyperbolas are all conic sections. Out of these conic sections, the circle and ellipse are the ones which define a closed curve.
Circles, ellipses, parabolas, and hyperbolas are called conic sections because they can be obtained as a intersection of a plane with a double- napped circular cone. If the plane passes through vertex of the double-napped cone, then the intersection is a point, a pair of straight lines or a single line. These are called degenerate conic sections. Because they are sections of a cone or a cone shaped object.
Ellipse circle
You can find them in mountains, in balls, and in tables.
They are the shapes of the slices when you slice a cone. For example, when you slice it parallel to the base and look at the shape of the slice, you see the conic section known as a "circle". The others are the "ellipse", the "parabola", and the "hyperbola". Which one you get depends only on how you tilt the knife when you slice the cone.