What kind of conic section

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. These distances are displayed as orange lines for each conic section in the following diagram. Parts of conic sections : The three conic sections with foci and directrices labeled. 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. 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. Ellipse: The sum of the distances from any point on the ellipse to the foci is constant. 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. 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. 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. They could follow ellipses, parabolas, or hyperbolas, depending on their properties. It can be thought of as a measure of how much the conic section deviates from being circular. The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix.

This property can be used as a general definition for conic sections. The eccentricity of a circle is zero. Note that two conic sections are similar identically shaped if and only if they have the same eccentricity. Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix. In the next figure, each type of conic section is graphed with a focus and directrix.

The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix.

Each shape also has a degenerate form. This figure shows how the conic sections, in light blue, are the result of a plane intersecting a cone. Image 1 shows a parabola, image 2 shows a circle bottom and an ellipse top , and image 3 shows a hyperbola. A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane. Every parabola has certain features:.

As a direct result of having the same eccentricity, all parabolas are similar, meaning that any parabola can be transformed into any other with a change of position and scaling. The degenerate case of a parabola is when the plane just barely touches the outside surface of the cone, meaning that it is tangent to the cone. This creates a straight line intersection out of the cone's diagonal. Non-degenerate parabolas can be represented with quadratic functions such as. A circle is formed when the plane is parallel to the base of the cone.

Its intersection with the cone is therefore a set of points equidistant from a common point the central axis of the cone , which meets the definition of a circle.

All circles have certain features:. Thus, like the parabola, all circles are similar and can be transformed into one another. On a coordinate plane, the general form of the equation of the circle is. None of the intersections will pass through the vertices of the cone. If the right circular cone is cut by a plane perpendicular to the axis of the cone, the intersection is a circle. If the plane intersects one of the pieces of the cone and its axis but is not perpendicular to the axis, the intersection will be an ellipse.

To generate a parabola, the intersecting plane must be parallel to one side of the cone and it should intersect one piece of the double cone. And finally, to generate a hyperbola the plane intersects both pieces of the cone.

For this, the slope of the intersecting plane should be greater than that of the cone. As we change the values of some of the constants, the shape of the corresponding conic will also change. It is important to know the differences in the equations to help quickly identify the type of conic that is represented by a given equation.

Center is h , k. Length of major axis is 2 a. Length of minor axis is 2 b. Distance between the vertices is 2 a. Distance between the foci is 2 c.