CONIC SECTIONS VIDEO: Animation of a moving plane (purple) intersecting a cone (green) and generating conic sections (bright pink). Depending on the angle of the plane, the intersection can be a circle, ellipse, parabola or hyperbola.
A circle (shown here in bright pink) is formed when a plane intersects a cone parallel to the base of the cone. A circle is a special case of an ellipse.
An ellipse is formed when a plane intersects a cone but fails to cross the base of the cone i.e. the ellipse is a closed figure unlike a parabola or an hyperbola. An ellipse is like a flattened circle and is a very important figure since it describes the orbits of the planets around the sun. When the interesecting plane lies parallel to the base of the cone then the ellipse becomes equal to a circle (a circle is a special case of an ellipse).
A parabola is formed when a plane intersects a cone and that plane lies parallel to the edge of that cone. Because the plane lies parallel to the side of the cone it always passes through the base of the cone (shown in blue) and so the figure remains open ended (unlike an ellipse, which is a related closed figure). Parabolas are very important figures since they describe the trajectory (flight path) of an object that is thrown and is pulled by gravity. Such an object is called "ballistic". Parabolic reflectors are concave mirrors with a parabolic profile (cross-section) that focus incoming rays to a point (focus). These are very important in astronomy where they are used in reflecting telescopes.
An hyperbola is formed when a plane intersects a cone at an angle less the the cone side makes with the cone axis. It is a double, open ended curve.