Equation of a Chord with Given Midpoint

In the study of circles, a chord is a line segment with both endpoints on the circle. The equation of a chord can be derived if the midpoint of the chord is known. This is particularly useful in coordinate geometry, where we often work with equations to represent geometric figures.

Understanding the Chord of a Circle

A chord of a circle divides the circle into two segments - the major segment and the minor segment. The longest chord of a circle is the diameter, which passes through the center of the circle. Any other chord will be shorter than the diameter.

Equation of a Circle

Before we derive the equation of a chord, let's recall the equation of a circle with center at $(h, k)$ and radius $r$:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Deriving the Equation of a Chord with a Given Midpoint

Let's consider a circle with center at $(h, k)$ and radius $r$. Let $(x_1, y_1)$ be the midpoint of a chord of the circle. The equation of the chord can be found using the following steps:

1. Write the equation of the circle.
2. Find the slope of the radius that passes through the midpoint of the chord.
3. Use the fact that the chord is perpendicular to the radius at its midpoint.
4. Write the equation of the chord using the point-slope form of a line.

Step-by-Step Derivation

1. Equation of the circle: $(x - h)^2 + (y - k)^2 = r^2$.
2. Slope of the radius: The slope of the radius connecting the center $(h, k)$ to the midpoint $(x_1, y_1)$ is $\frac{y_1 - k}{x_1 - h}$.
3. Perpendicular slope: The chord is perpendicular to the radius, so its slope is the negative reciprocal of the radius' slope. Therefore, the slope of the chord, $m$, is $m = -\frac{x_1 - h}{y_1 - k}$.
4. Equation of the chord: Using the point-slope form, $y - y_1 = m(x - x_1)$, we get the equation of the chord:

$$ y - y_1 = -\frac{x_1 - h}{y_1 - k}(x - x_1) $$

Important Points and Differences

Aspect	Circle	Chord with Given Midpoint
Definition	A set of points equidistant from a common center.	A line segment with endpoints on the circle.
Equation	$(x - h)^2 + (y - k)^2 = r^2$	$y - y_1 = -\frac{x_1 - h}{y_1 - k}(x - x_1)$
Known Parameters	Center $(h, k)$, radius $r$	Midpoint $(x_1, y_1)$
Slope	Not applicable	$-\frac{x_1 - h}{y_1 - k}$ (perpendicular to radius)
Relationship to Radius	Radius is the distance from the center to any point on the circle.	Chord is perpendicular to the radius at its midpoint.

Examples

Example 1: Finding the Equation of a Chord

Given a circle with center at $(2, 3)$ and radius $5$, find the equation of the chord whose midpoint is at $(4, 6)$.

1. Equation of the circle: $(x - 2)^2 + (y - 3)^2 = 25$.
2. Slope of the radius: $\frac{6 - 3}{4 - 2} = \frac{3}{2}$.
3. Perpendicular slope: $-\frac{4 - 2}{6 - 3} = -\frac{2}{3}$.
4. Equation of the chord: $y - 6 = -\frac{2}{3}(x - 4)$, which simplifies to $y = -\frac{2}{3}x + 10$.

Example 2: Using the Equation of a Chord

Suppose you have a circle with center at $(0, 0)$ and radius $10$, and you know that the midpoint of a chord is at $(6, 8)$. What is the equation of this chord?

1. Equation of the circle: $x^2 + y^2 = 100$.
2. Slope of the radius: Since the center is at the origin, the slope of the radius is $\frac{8 - 0}{6 - 0} = \frac{4}{3}$.
3. Perpendicular slope: $-\frac{6 - 0}{8 - 0} = -\frac{3}{4}$.
4. Equation of the chord: $y - 8 = -\frac{3}{4}(x - 6)$, which simplifies to $y = -\frac{3}{4}x + 12.5$.

Understanding the equation of a chord with a given midpoint is a valuable tool in coordinate geometry, especially when dealing with problems involving circles and their properties.