Equation of Chord with Midpoints

When studying conic sections such as ellipses, the equation of a chord with a given midpoint is a common topic. A chord is a line segment whose endpoints lie on the curve. The midpoint of a chord is the point that divides the chord into two equal segments. Understanding the equation of a chord with a given midpoint is crucial for solving various geometric and algebraic problems related to conics.

General Equation of an Ellipse

Before we delve into the equation of a chord with midpoints, let's recall the general equation of an ellipse centered at the origin:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

where (a) is the semi-major axis and (b) is the semi-minor axis of the ellipse.

Equation of a Chord with a Given Midpoint

If a chord of the ellipse has a midpoint at ((x_1, y_1)), then the equation of the chord can be derived using the midpoint formula and the general equation of the ellipse.

The midpoint formula for a chord with endpoints ((x_2, y_2)) and ((x_3, y_3)) is given by:

$$ x_1 = \frac{x_2 + x_3}{2}, \quad y_1 = \frac{y_2 + y_3}{2} $$

Since the endpoints of the chord lie on the ellipse, they satisfy the ellipse's equation:

$$ \frac{x_2^2}{a^2} + \frac{y_2^2}{b^2} = 1, \quad \frac{x_3^2}{a^2} + \frac{y_3^2}{b^2} = 1 $$

By using the concept of conjugate points, we can derive the equation of the chord as:

$$ \frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1 $$

This is the equation of the chord with midpoint ((x_1, y_1)) for the ellipse.

Differences and Important Points

Let's summarize the differences and important points in a table:

Aspect	Chord with Given Midpoint	General Chord
Definition	A line segment with endpoints on the ellipse and a specified midpoint.	A line segment with endpoints on the ellipse, but no specific midpoint is given.
Equation	( \frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1 )	Varies depending on the endpoints and the ellipse's equation.
Midpoint	Given ((x_1, y_1))	Calculated using the endpoints of the chord.
Use	Solving problems where the midpoint is known or needs to be found.	General analysis of chords on an ellipse.

Examples

Example 1: Finding the Equation of a Chord

Given an ellipse with (a = 5) and (b = 3), find the equation of the chord whose midpoint is ((2, 1)).

Solution:

Using the formula for the equation of the chord with midpoint ((x_1, y_1)), we have:

$$ \frac{x \cdot 2}{5^2} + \frac{y \cdot 1}{3^2} = 1 $$

Simplifying, we get:

$$ \frac{2x}{25} + \frac{y}{9} = 1 $$

Multiplying through by the common denominator (225), we obtain:

$$ 18x + 25y = 225 $$

This is the equation of the chord with midpoint ((2, 1)) on the given ellipse.

Example 2: Verifying a Point on the Chord

Verify if the point ((5, 4)) lies on the chord of the ellipse (\frac{x^2}{25} + \frac{y^2}{9} = 1) with midpoint ((2, 1)).

Solution:

The equation of the chord is (18x + 25y = 225). Substituting the point ((5, 4)) into the equation:

$$ 18 \cdot 5 + 25 \cdot 4 = 90 + 100 = 190 $$

Since (190 \neq 225), the point ((5, 4)) does not lie on the chord.

Understanding the equation of a chord with a given midpoint is essential for solving problems related to the geometry of ellipses. It allows us to find the specific line segment that connects two points on the ellipse and passes through a particular midpoint.