# 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:

$\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}=1$

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 x-axis focus at $\left(2,0\right)$. The corresponding point on the hyperbola would be $\left(c,0\right)$, 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 $\left(\sqrt{13},0\right)$. Let's choose another point on the hyperbola, say $\left(-2,0\right)$, as the other endpoint.

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

$\text{Length}=\sqrt{\left(\sqrt{13}-\left(-2\right){\right)}^{2}+\left(0-0{\right)}^{2}}$

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 $\left(-3,2\right)$ and $\left(1,2\right)$, as the endpoints of the chord.

Calculate the length of this chord using the distance formula:

$\text{Length}=\sqrt{\left(1-\left(-3\right){\right)}^{2}+\left(2-2{\right)}^{2}}$

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 $\left(-2,1\right)$ and $\left(2,1\right)$. The conjugate chord parallel to the conjugate axis would have endpoints $\left(1,2\right)$ and $\left(1,-2\right)$.

Calculate the lengths of these conjugate chords.

4. Parametric Representation: The parametric equations for the hyperbola are: $x\left(t\right)=2\mathrm{sec}\left(t\right)$ $y\left(t\right)=3\mathrm{tan}\left(t\right)$

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