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Which conic section is formed when a plane cuts both parts of a double cone?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

The conic section formed when a plane cuts both parts (or nappes) of a double cone is a hyperbola.

A hyperbola is defined as a two-branched open curve. In analytic geometry, it can be described as the locus of a point moving such that the ratio of its distance from a fixed point (the focus) to its distance from a fixed line (the directrix) is a constant greater than one. Because of its symmetry, a hyperbola possesses two foci. A degenerate case, consisting of two intersecting lines, occurs if the plane passes through the apex of the cone.12