Chords of an Ellipse

Chords of an Ellipse

Definition:

  • Chord: In geometry, a chord is a straight line segment that joins two points on the curve (in this case, an ellipse) but does not necessarily pass through the center of the curve.

Properties and Concepts:

  1. Length of Chords:

    • Chords of an ellipse vary in length, with the longest chord being the major axis, and the shortest being the minor axis.
  2. Secant and Tangent Lines:

    • A chord that passes through the center of the ellipse is called a diameter.
    • A chord that touches the ellipse at only one point is called a tangent.
  3. Parallel Chords:

    • Parallel chords of an ellipse have equal lengths, and their midpoints lie on the minor axis.
  4. Chord Bisected by Major or Minor Axis:

    • A chord that is bisected by the major axis is parallel to the minor axis and vice versa.

Equation of a Chord

Definition:

  • The equation of a chord passing through two points (x1,y1) and (x2,y2) on an ellipse is derived using the coordinates of the endpoints and the general equation of an ellipse.

Steps to Find the Equation:

  1. General Equation of Ellipse:

    • The standard equation of an ellipse is (xh)2a2+(yk)2b2=1.
  2. Coordinates of Endpoints:

    • Given two points (x1,y1) and (x2,y2) lying on the ellipse, substitute these values into the standard equation to form a system of equations.
  3. Solving the System of Equations:

    • Solve the equations obtained from substituting the coordinates of endpoints into the general equation to find the parameters (a,b,h,k).
  4. Forming the Chord Equation:

    • Once the parameters (a,b,h,k) are known, plug them into the general equation to form the equation of the chord.

Diameter of an Ellipse

Definition:

  • Diameter: A special chord of an ellipse that passes through the center, having endpoints on the ellipse. It is the longest chord of the ellipse.

Characteristics:

  • Properties: The diameter is perpendicular to the major axis and passes through the center, bisecting the ellipse into two equal halves.

Equation of the Diameter:

  • Standard Form: For an ellipse with center (h,k), the equation of the diameter along the x-axis is x=h and along the y-axis is y=k.

Conjugate Diameters of an Ellipse

Definition:

  • Conjugate diameters of an ellipse are pairs of diameters that are perpendicular to each other. When two diameters are perpendicular, they are referred to as conjugate diameters.

Properties:

  1. Perpendicularity:

    • Conjugate diameters are always perpendicular to each other. This characteristic distinguishes them from other pairs of diameters on the ellipse.
  2. Interchangeability:

    • If one of the conjugate diameters is a major axis, the other is a minor axis and vice versa. They are interchangeable in terms of their lengths and orientation.
  3. Product of Lengths:

The product of the lengths of conjugate diameters is constant and is equal to the square of the length of the semi-major axis multiplied by the square of the length of the semi-minor axis.

   Product of lengths of conjugate diameters=a2×b2 

   Where a is the length of the semi-major axis and b is the length of the semi-minor axis.

4. Geometric Construction:

Given an ellipse, finding one conjugate diameter allows the construction of the other by drawing a line perpendicular to it and passing through the center of the ellipse.

5. Orthogonality in Orthogonal Projections:

Conjugate diameters maintain their perpendicularity even in the process of orthogonal projection, aiding in various geometric transformations and applications.