Equation of Chord with Midpoints

When studying conic sections such as hyperbolas, the equation of a chord with given midpoints is an important concept. A chord is a line segment whose endpoints lie on the curve, and the midpoint is the point that divides the chord into two equal segments. Understanding how to derive and use the equation of a chord with a given midpoint is crucial for solving various problems in coordinate geometry.

General Equation of a Chord

For a conic section given by the general quadratic equation:

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, $$

the chord with midpoint $(x_1, y_1)$ can be represented by the equation:

$$ T = S_1, $$

where $T$ is the expression obtained by replacing $x^2$ with $xx_1$, $y^2$ with $yy_1$, and $xy$ with $\frac{1}{2}(xy_1 + yx_1)$ in the given conic equation, and $S_1$ is the value of the original conic equation when $(x, y)$ is replaced by $(x_1, y_1)$.

Equation of Chord with Midpoints for a Hyperbola

For a hyperbola defined by the equation:

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

the equation of the chord with midpoint $(x_1, y_1)$ is given by:

$$ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1. $$

This equation is derived by applying the midpoint formula to the hyperbola's equation.

Table of Differences and Important Points

Feature	Chord of Hyperbola	Chord of Circle	Chord of Parabola	Chord of Ellipse
Equation Form	$\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$	$xx_1 + yy_1 = r^2$	$yy_1 = 2p(x + x_1)$	$\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$
Midpoint Coordinates	$(x_1, y_1)$	$(x_1, y_1)$	$(x_1, y_1)$	$(x_1, y_1)$
Conic Section	Hyperbola	Circle	Parabola	Ellipse
Orientation	Depends on $a^2$ and $b^2$ values	Centered at origin	Axis of symmetry	Major and minor axes
Asymptotes	Yes	No	No	No

Examples

Example 1: Hyperbola Chord

Given the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ and the midpoint of the chord $(2, -3)$, find the equation of the chord.

Solution:

Using the formula for the chord with midpoint $(x_1, y_1)$:

$$ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1, $$

we substitute $a^2 = 16$, $b^2 = 9$, $x_1 = 2$, and $y_1 = -3$:

$$ \frac{x \cdot 2}{16} - \frac{y \cdot (-3)}{9} = 1, $$

which simplifies to:

$$ \frac{x}{8} + \frac{y}{3} = 1. $$

Therefore, the equation of the chord is:

$$ \frac{x}{8} + \frac{y}{3} = 1. $$

Example 2: Circle Chord

For a circle with radius $r = 5$ and a chord with midpoint $(3, 4)$, find the equation of the chord.

Solution:

The general equation for a chord with midpoint $(x_1, y_1)$ in a circle is:

$$ xx_1 + yy_1 = r^2, $$

Substituting the given values:

$$ x \cdot 3 + y \cdot 4 = 5^2, $$

which simplifies to:

$$ 3x + 4y = 25. $$

Therefore, the equation of the chord is:

$$ 3x + 4y = 25. $$

Understanding the equation of a chord with midpoints for different conic sections is essential for solving problems related to geometry and conic sections. The above examples illustrate how to apply the formulas to find the equations of chords for a hyperbola and a circle.