Defining Conic Sections
A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic sections are the hyperbola, the parabola, and the ellipse. The circle is type of ellipse, and is sometimes considered to be a fourth type of conic section.
Conic sections can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. One nappe is what most people mean by “cone,” and has the shape of a party hat.
Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the
A cone and conic sections
The nappes and the four conic sections. Each conic is determined by the angle the plane makes with the axis of the cone.
Common Parts of Conic Sections
While each type of conic section looks very different, they have some features in common. For example, each type has at least one focus and directrix.
A focus is a point about which the conic section is constructed. In other words, it is a point about which rays reflected from the curve converge. A parabola has one focus about which the shape is constructed; an ellipse and hyperbola have two.
A directrix is a line used to construct and define a conic section. The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.
These properties that the conic sections share are often presented as the following definition, which will be developed further in the following section. A conic section is the locus of points
Parts of conic sections
The three conic sections with foci and directrices labeled.
Each type of conic section is described in greater detail below.
Parabola
A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola.
In the next figure, four parabolas are graphed as they appear on the coordinate plane. They may open up, down, to the left, or to the right.
Four parabolas, opening in various directions
The vertex lies at the midpoint between the directrix and the focus.
Ellipses
An ellipse is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. In the case of an ellipse, there are two foci, and two directrices.
In the next figure, a typical ellipse is graphed as it appears on the coordinate plane.
Ellipse
The sum of the distances from any point on the ellipse to the foci is constant.
Hyperbolas
A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes.
A graph of a typical hyperbola appears in the next figure.
Hyperbola
The difference of the distances from any point on the ellipse to the foci is constant. The transverse axis is also called the major axis, and the conjugate axis is also called the minor axis.
Applications of Conic Sections
Conic sections are used in many fields of study, particularly to describe shapes. For example, they are used in astronomy to describe the shapes of the orbits of objects in space. Two massive objects in space that interact according to Newton's law of universal gravitation can move in orbits that are in the shape of conic sections. They could follow ellipses, parabolas, or hyperbolas, depending on their properties.