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:

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.

Transverse Axis Chord:
 A chord that is parallel to the transverse axis of the hyperbola.

Conjugate Axis Chord:
 A chord that is parallel to the conjugate axis of the hyperbola.
III. Key Properties:

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

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 nonfocal chord.
 The position of a chord in relation to the foci can provide insights into its nature:
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:
$\frac{{x}^{2}}{4}\frac{{y}^{2}}{9}=1$
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 xaxis 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, $c=\sqrt{{a}^{2}+{b}^{2}}=\sqrt{4+9}=\sqrt{13}$.
So, one endpoint of the focal chord is $(\sqrt{13},0)$. 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:
$\text{Length}=\sqrt{(\sqrt{13}(2){)}^{2}+(00{)}^{2}}$
Calculate the length to find the value.
2. Nonfocal Chord: Let's consider a nonfocal 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:
$\text{Length}=\sqrt{(1(3){)}^{2}+(22{)}^{2}}$
This nonfocal 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: $x(t)=2\mathrm{sec}(t)$ $y(t)=3\mathrm{tan}(t)$
Choose a parameter $t$ and find two points on the hyperbola. Use these points to form a chord and calculate its length.