conic section

Examples of conic section in the following topics:

• Ellipses as Conic Sections

• Hyperbolas as Conic Sections

• Eccentricity

  • The eccentricity, denoted $e$, is a parameter associated with every conic section.
  • The value of $e$ is constant for any conic section.
  • This property can be used as a general definition for conic sections.
  • The value of $e$ can be used to determine the type of conic section as well:
  • Explain how the eccentricity of a conic section describes its behavior

• What Are Conic Sections?

  • A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
  • A focus is a point about which the conic section is constructed.
  • Each type of conic section is described in greater detail below.
  • The nappes and the four conic sections.
  • Describe the parts of a conic section and how conic sections can be thought of as cross-sections of a double-cone

• Conics in Polar Coordinates

  • Polar coordinates allow conic sections to be expressed in an elegant way.
  • With this definition, we may now define a conic in terms of the directrix: $x=±p$, the eccentricity $e$, and the angle $\theta$.
  • Thus, each conic may be written as a polar equation in terms of $r$ and $\theta$.
  • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:

• Types of Conic Sections

  • Conic sections are formed by the intersection of a plane with a cone, and their properties depend on how this intersection occurs.
  • Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone.
  • There is a property of all conic sections called eccentricity, which takes the form of a numerical parameter $e$.
  • The four conic section shapes each have different values of $e$.
  • This figure shows how the conic sections, in light blue, are the result of a plane intersecting a cone.

• Parabolas As Conic Sections

  • Parabolas are one of the four shapes known as conic sections, and they have many important real world applications.
  • In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.
  • In the diagram showing the parabolic conic section, a red line is drawn from the center of that circle to the axis of symmetry, so that a right angle is formed.
  • Describe the parts of a parabola as parts of a conic section

• Parts of an Ellipse

  • Ellipses are one of the types of conic sections.
  • An ellipse is a conic section, formed by the intersection of a plane with a right circular cone.
  • All conic sections have an eccentricity value, denoted $e$.

• Nonlinear Systems of Equations and Problem-Solving

  • A conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane.
  • The four types of conic section are the hyperbola, the parabola, the ellipse, and the circle, though the circle can be considered to be a special case of the ellipse.
  • The type of a conic corresponds to its eccentricity.
  • Conics with eccentricity less than $1$ are ellipses, conics with eccentricity equal to $1$ are parabolas, and conics with eccentricity greater than $1$ are hyperbolas.
  • Nonlinear systems of equations, such as conic sections, include at least one equation that is nonlinear.

• Converting the Conic Equation of a Parabola to Standard Form