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How are conic sections formed by slicing a cone with a plane?

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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).

Conic sections are curves created by the intersection of a plane and a right circular cone. The specific type of curve produced depends on the angle at which the plane slices through the cone.1

The four primary conic sections are:
* Circle: Formed when the plane cuts horizontally across the cone.
* Ellipse: Formed when the plane slices through the cone at an angle, creating a closed curve.
* Parabola: Formed when the plane is parallel to the side (slant height) of the cone.
* Hyperbola: Formed when the plane cuts through both nappes of the cone, resulting in two separate curves.

Additionally, there are degenerate cases where the plane passes through the cone's apex. These intersections produce a single point, a single straight line, or two intersecting straight lines.1

Would you like to learn more about the algebraic equations used to define these curves in analytic geometry?