ellipse

(noun)

The conic section formed by the plane being at an angle to the base of the cone.

Related Terms

  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • hyperbola
  • focal point
  • vertices
  • asymptote
  • vertex

(noun)

One of the conic sections.

Related Terms

  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • hyperbola
  • focal point
  • vertices
  • asymptote
  • vertex

(noun)

A closed curve, the locus of a point such that the sum of the distances from that point to two other fixed points (called the foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone.

Related Terms

  • ci
  • center
  • cente
  • Asymptote
  • degenerate
  • circle
  • focus
  • eccentricity
  • Parabola
  • hyperbola
  • focal point
  • vertices
  • asymptote
  • vertex

Examples of ellipse in the following topics:

  • Ellipses as Conic Sections

  • Parts of an Ellipse

    • Ellipses are one of the types of conic sections.
    • The standard form for the equation of the ellipse is:
    • if the ellipse is oriented vertically.
    • For a vertical ellipse, the association is reversed.
    • An eccentricity of $1$ is a parabola, not an ellipse.
  • Ellipses

    • And the resulting shape will be an ellipse.
    • How often do ellipses come up in real life?
    • The sun is at one focus of the ellipse (not at the center).
    • If a>b, the ellipse is horizontal.
    • If a, the the ellipse is vertical.
  • Introduction to Ellipses

    • An ellipse is one of the shapes called conic sections, which is formed by the intersection of a plane with a right circular cone.
    • The general equation of an ellipse centered at $\left(h,k\right)$ is:
    • which is exactly the equation of a horizontal ellipse centered at the origin.
    • An ellipse is a conic section, formed by the intersection of a plane with a right circular cone.
    • Connect the equation for an ellipse to the equation for a circle with stretching factors
  • What Are Conic Sections?

    • The three types of conic sections are the hyperbola, the parabola, and the ellipse.
    • 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.
    • They could follow ellipses, parabolas, or hyperbolas, depending on their properties.
    • The sum of the distances from any point on the ellipse to the foci is constant.
  • Types of Conic Sections

    • The definition of an ellipse includes being parallel to the base of the cone as well, so all circles are a special case of the ellipse.
    • Ellipses have these features:
    • A major axis, which is the longest width across the ellipse
    • A minor axis, which is the shortest width across the ellipse
    • Ellipses can have a range of eccentricity values: $0 \leq e < 1$.
  • Eccentricity

    • Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix.
    • For an ellipse, the eccentricity is less than $1$.
  • Applications of Circles and Ellipses

    • Circles and ellipses are encountered in everyday life, and knowing how to solve their equations is useful in many situations.
    • Ellipses are less common.
    • One example is the orbits of planets, but you should be able to find the area of a circle or an ellipse, or the circumference of a circle, based on information given to you in a problem.
    • Circles and ellipses are examples of conic sections, which are curves formed by the intersection of a plane with a cone.
    • This almost looks like an ellipse in standard form, doesn't it?
  • Nonlinear Systems of Equations and Problem-Solving

    • 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.
    • 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.
    • In the focus-directrix definition of a conic, the circle is a limiting case of the ellipse with an eccentricity of $0$.
  • Standard Equations of Hyperbolas

    • Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does.
    • A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices.
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