Ellipse

Ellipse: An Overview

Definition:

  • Ellipse: An ellipse is a type of conic section formed by the intersection of a cone and a plane. It's defined as the set of all points in a plane, the sum of whose distances from two fixed points (the foci) is constant.

Key Elements:

  1. Foci (Focus Singular): Two fixed points inside the ellipse, denoted as F1 and F2. The sum of distances from any point on the ellipse to these foci is constant.

  2. Major and Minor Axes:

    • Major Axis: The longest diameter passing through the foci, endpoints called vertices.
    • Minor Axis: The shortest diameter perpendicular to the major axis, also with endpoints on the ellipse.
  3. Center: The midpoint of the major and minor axes, denoted as (h, k).

Standard Equation:

The standard form of the equation for an ellipse centered at (h,k) with major axis of length 2a and minor axis of length 2b is:

(xh)2a2+(yk)2b2=1

Rotated Ellipse: An ellipse that is tilted or rotated at an angle θ with respect to the x-axis. Its equation can be derived through rotation formulas.

Eccentricity:

  • Eccentricity (e): A measure of how much the ellipse deviates from a circle. It's calculated as e=ca, where c is the distance from the center to a focus and a is half the length of the major axis.

  • For an ellipse, 0<e<1:

    • e=0 for a circle.
    • 0<e<1 for an ellipse.
    • e=1 for a parabola (an extreme case of an ellipse).
    • e>1 for a hyperbola.

Properties and Characteristics:

  1. Symmetry: Ellipses are symmetric about their major and minor axes.

  2. Foci-Distance Relationship: The distance between the foci is constant and equal to 2c, where c is the distance between the center and each focus.
  3. Vertex: The points where the ellipse intersects the major and minor axes are called vertices.
  4. Area: The area of an ellipse is given by A=πab, where a and b are the semi-major and semi-minor axes, respectively
  5. Relationship between Axes and Eccentricity:

    • The relationship b=a1e2 holds for the lengths of the major and minor axes.
  6. Parametric Equations:

    • Parametric equations of an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b are: x=h+acos(θ) y=k+bsin(θ)
  7. Application in Orbits and Optics:

    • Orbits of celestial bodies often follow elliptical paths.
    • In optics, ellipses are used in designing lenses, telescopes, and satellite dishes.