conic section

(noun)

Any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola, and hyperbola.

Related Terms

  • trilateration
  • directrix
  • nappe
  • focus
  • nonlinear
  • hyperbola
  • asymptote
  • system of equations
  • vertex
  • locus

(noun)

Any curve formed by the intersection of a plane with a cone of two nappes.

Related Terms

  • trilateration
  • directrix
  • nappe
  • focus
  • nonlinear
  • hyperbola
  • asymptote
  • system of equations
  • vertex
  • locus

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
  • Conic Sections

    • In mathematics, a conic section (or just "conic") is a curve obtained from the intersection of a cone (more precisely, a right circular conical surface) with a plane.
    • Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse.
    • In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section—though it may be degenerate—and all conic sections arise in this way.
    • There are three types of conic sections: 1.Parabola; 2.
    • Identify conic sections as curves obtained from the intersection of a cone with a plane
  • 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
  • Conic Sections in Polar Coordinates

    • Conic sections are sections of cones and can be represented by polar coordinates.
    • In mathematics, 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.
    • Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse.
    • The circle is a special case of the ellipse, and is of such sufficient interest in its own right that it is sometimes called the fourth type of conic section.
    • In polar coordinates, a conic section with one focus at the origin is given by the following equation:
  • 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
  • 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.
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