The definition of a circle, in the sense of it being a conic section, is when a right cone is intersected with a plane that is parallel with the base of the cone. We can also say that the angle of the plane is 90 degrees or perpendicular to the center axis of the cone. A circle is formed when you have all points equidistant from one point, the center.
An ellipse is formed when a right cone is intersected with a plane at an angle between the degree of the cone and 90 degrees. An ellipse can be drawn from two foci. Mathwords defines an ellipse as the following: "For two given points, the foci, an ellipse is the locus of points such that the sum of the distance to each focus is constant." (1) The derivation for the standard form of an ellipse can be found below.
A parabola is formed when a right cone is intersected with a plane at the exact angle of the cone. The derivation for the standard form of a parabola is found below.
A hyperbola is formed when a double right cone is intersected with a plane at an angle between the angle of the cone and 0 degrees. The derivation for the standard form of a hyperbola is found below.
