# 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 $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ 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 $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}+\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$.
2. Coordinates of Endpoints:

• Given two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ 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 $\left(h,k\right)$, 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.

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.