Chords of a Hyperbola

I. Introduction to Chords:

  • In geometry, a chord is a line segment with both endpoints on the curve of a figure.
  • For hyperbolas, chords play a significant role in understanding the relationships between points on the curve.

II. Definitions:

  1. Chord of a Hyperbola:

    • A chord of a hyperbola is a line segment joining two points on the hyperbola.
    • It lies entirely within the curve and is a straight line passing through the hyperbolic plane.
  2. Transverse Axis Chord:

    • A chord that is parallel to the transverse axis of the hyperbola.
  3. Conjugate Axis Chord:

    • A chord that is parallel to the conjugate axis of the hyperbola.

III. Key Properties:

  1. Length of a Chord:

    • The length of a chord can be calculated using the distance formula between its endpoints.
  2. Position of the Chord:

    • The position of a chord in relation to the foci can provide insights into its nature:
      • If the chord passes through a focus, it is called a focal chord.
      • If it does not pass through any focus, it is a non-focal chord.

IV. Focal Chords:

  • A focal chord of a hyperbola passes through one of the foci.
  • The product of the distances from the foci to the endpoints of a focal chord is constant.

V. Parametric Representation:

  • Chords can be parametrically represented using the parametric equations of the hyperbola.

VI. Conjugate Chords:

  • Conjugate chords are pairs of chords of a hyperbola that are parallel to each other.
  • The product of the lengths of corresponding conjugate chords is constant.

Example: 

Consider the hyperbola given by the equation:

x24y29=1

Let's explore a few chords on this hyperbola and analyze their properties.

1. Focal Chord: A focal chord is a chord that passes through one of the foci. Let's consider a focal chord passing through the positive x-axis focus at (2,0). The corresponding point on the hyperbola would be (c,0), where c is the distance from the center to the focus.

For this hyperbola, c=a2+b2=4+9=13.

So, one endpoint of the focal chord is (13,0). Let's choose another point on the hyperbola, say (2,0), as the other endpoint.

The length of the focal chord is given by the distance formula:

Length=(13(2))2+(00)2

Calculate the length to find the value.

2. Non-focal Chord: Let's consider a non-focal chord that does not pass through either focus. Choose two arbitrary points on the hyperbola, say (3,2) and (1,2), as the endpoints of the chord.

Calculate the length of this chord using the distance formula:

Length=(1(3))2+(22)2

This non-focal chord is entirely within the hyperbola but doesn't pass through the foci.

3. Conjugate Chords: Conjugate chords are parallel chords. Choose two parallel chords, one parallel to the transverse axis and the other parallel to the conjugate axis.

For instance, consider a chord parallel to the transverse axis with endpoints (2,1) and (2,1). The conjugate chord parallel to the conjugate axis would have endpoints (1,2) and (1,2).

Calculate the lengths of these conjugate chords.

4. Parametric Representation: The parametric equations for the hyperbola are: x(t)=2sec(t) y(t)=3tan(t)

Choose a parameter t and find two points on the hyperbola. Use these points to form a chord and calculate its length.