Equation of Chord with Midpoints

When studying the geometry of circles, one of the concepts that often comes up is the equation of a chord whose midpoint is known. A chord of a circle is a straight line segment whose endpoints both lie on the circle. The midpoint of the chord is the point that divides the chord into two equal segments.

Derivation of the Equation

Let's consider a circle with the equation:

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

where $(h, k)$ is the center of the circle and $r$ is the radius.

Suppose we have a chord of this circle, and we know the coordinates of its midpoint, say $(x_1, y_1)$. To find the equation of this chord, we can use the fact that the perpendicular from the center of the circle to the chord is a bisector and will pass through the midpoint of the chord.

The slope of the line passing through the center $(h, k)$ and the midpoint $(x_1, y_1)$ is:

$$ m = \frac{y_1 - k}{x_1 - h} $$

The slope of the chord will be the negative reciprocal of this slope (since the chord is perpendicular to the radius at the midpoint), which is:

$$ m_{chord} = -\frac{x_1 - h}{y_1 - k} $$

Using the point-slope form of the line equation, the equation of the chord is:

$$ y - y_1 = m_{chord}(x - x_1) $$

Substituting the value of $m_{chord}$, we get:

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

This can be rearranged to:

$$ (y_1 - k)(y - y_1) = -(x_1 - h)(x - x_1) $$

Expanding and simplifying, we obtain the equation of the chord:

$$ (y_1 - k)y + (x_1 - h)x = (y_1 - k)y_1 + (x_1 - h)x_1 $$

Important Points and Differences

Aspect	Description
Equation of Circle	$(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.
Midpoint of Chord	The point that divides the chord into two equal parts.
Slope of Radius	$m = \frac{y_1 - k}{x_1 - h}$ where $(x_1, y_1)$ is the midpoint of the chord.
Slope of Chord	$m_{chord} = -\frac{x_1 - h}{y_1 - k}$ which is the negative reciprocal of the slope of the radius.
Equation of Chord	$(y_1 - k)y + (x_1 - h)x = (y_1 - k)y_1 + (x_1 - h)x_1$

Examples

Example 1: Find the Equation of the Chord

Consider a circle with the equation $(x - 3)^2 + (y + 2)^2 = 25$ and a chord with a midpoint at $(4, -1)$.

1. Find the slope of the radius to the midpoint: $$ m = \frac{-1 + 2}{4 - 3} = 1 $$

2. Find the slope of the chord: $$ m_{chord} = -\frac{4 - 3}{-1 + 2} = -1 $$

3. Use the point-slope form to find the equation of the chord: $$ y + 1 = -1(x - 4) $$ $$ y = -x + 3 $$

Example 2: Find the Equation of the Chord with a Different Midpoint

Consider the same circle $(x - 3)^2 + (y + 2)^2 = 25$ and a chord with a midpoint at $(5, 0)$.

1. Find the slope of the radius to the midpoint: $$ m = \frac{0 + 2}{5 - 3} = 1 $$

2. Find the slope of the chord: $$ m_{chord} = -\frac{5 - 3}{0 + 2} = -1 $$

3. Use the point-slope form to find the equation of the chord: $$ y - 0 = -1(x - 5) $$ $$ y = -x + 5 $$

In both examples, we used the known midpoint of the chord to derive the equation of the chord itself. This method is widely applicable and can be used for any circle and any chord, provided the midpoint is known.