A chord of a parabola is a straight line segment that connects any two distinct points on the parabola. Unlike a unique tangent or normal, there isn't a single "equation of the chord" for a parabola. Instead, the equation varies based on the specific information provided, such as the coordinates of its endpoints, its midpoint, or if it passes through a special point like the focus.
To understand the equations, we will primarily consider the standard form of a parabola: $y^2 = 4ax$.
General Equation of a Chord Connecting Two Points
To define a specific chord, we typically need two points that lie on the parabola.
Using Cartesian Coordinates
If two points on the parabola $y^2 = 4ax$ are $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the equation of the chord connecting them is the equation of the line passing through these two points.
The general equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
Since $P_1$ and $P_2$ lie on $y^2 = 4ax$, we know $y_1^2 = 4ax_1$ and $y_2^2 = 4ax_2$. Using these relationships, the equation can be simplified to:
$\mathbf{y(y_1 + y_2) = 4ax + y_1y_2}$
This provides a direct method to find the chord's equation when two Cartesian points on the parabola are known.
Using Parametric Coordinates
For the parabola $y^2 = 4ax$, any point on the parabola can be conveniently represented using parametric coordinates as $(at^2, 2at)$. Let the two endpoints of the chord be $P_1(at_1^2, 2at_1)$ and $P_2(at_2^2, 2at_2)$, where $t_1$ and $t_2$ are parameters for the respective points.
The slope of the chord connecting $P_1$ and $P_2$ is:
$m = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_1 + t_2}$ (assuming $t_1 \neq t_2$)
Using the point-slope form $y - y_1 = m(x - x_1)$ with $P_1(at_1^2, 2at_1)$ and the calculated slope:
$y - 2at_1 = \frac{2}{t_1 + t_2}(x - at_1^2)$
After algebraic rearrangement, the general equation of a chord connecting two points $P_1(t_1)$ and $P_2(t_2)$ on the parabola $y^2 = 4ax$ is:
$\mathbf{2x - (t_1 + t_2)y + 2at_1t_2 = 0}$
This parametric form is frequently used in coordinate geometry for its simplicity and ease of manipulation.
The Focal Chord: A Special Case
A particularly significant type of chord is the focal chord. A focal chord of the parabola $y^2 = 4ax$ is defined as any chord that passes through the parabola's focus $(a, 0)$.
For the general chord equation in parametric form, $2x - (t_1 + t_2)y + 2at_1t_2 = 0$, to represent a focal chord, it must pass through the focus $(a, 0)$. Substituting $x=a$ and $y=0$ into the equation:
$2(a) - (t_1 + t_2)(0) + 2at_1t_2 = 0$
$2a + 2at_1t_2 = 0$
$2a(1 + t_1t_2) = 0$
Since $a \neq 0$ for a parabola, it must be that $1 + t_1t_2 = 0$.
Therefore, the essential condition for a chord connecting $P_1(at_1^2, 2at_1)$ and $P_2(at_2^2, 2at_2)$ to be a focal chord is:
$\mathbf{t_1t_2 = -1}$
When this condition is met, the equation of the focal chord simplifies to:
$2x - (t_1 + t_2)y + 2a(-1) = 0$
$\mathbf{2x - (t_1 + t_2)y - 2a = 0}$
This equation can also be expressed in terms of a single parameter, $t_1$, by substituting $t_2 = -1/t_1$:
$2x - \left(t_1 - \frac{1}{t_1}\right)y - 2a = 0$.
Equation of a Chord with a Given Midpoint
When a chord is defined by its midpoint rather than its endpoints, a different formula is used. If $M(x_M, y_M)$ is the midpoint of a chord of the parabola $y^2 = 4ax$, the equation of that chord is given by the formula $T=S_1$, where:
- $T$ represents the equation of the tangent at $(x_M, y_M)$ (but used for the chord): $yy_M - 2a(x + x_M)$
- $S_1$ represents the value of the parabola's equation $y^2 - 4ax$ at the midpoint: $y_M^2 - 4ax_M$
Therefore, the equation of the chord with midpoint $M(x_M, y_M)$ is:
$\mathbf{yy_M - 2a(x + x_M) = y_M^2 - 4ax_M}$
This is a powerful formula for directly finding the equation of a chord when its midpoint is known.
Summary of Chord Equations for Parabola $y^2 = 4ax$
The following table summarizes the different ways to express the equation of a chord depending on the given information for the standard parabola $y^2=4ax$.
| Type of Chord | Given Information | Equation |
|---|---|---|
| General Chord | Two Cartesian points $(x_1, y_1)$ and $(x_2, y_2)$ | $y(y_1 + y_2) = 4ax + y_1y_2$ |
| General Chord | Two parametric points $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$ | $2x - (t_1 + t_2)y + 2at_1t_2 = 0$ |
| Focal Chord | Passing through focus $(a,0)$ (parametric form) | $2x - (t_1 + t_2)y - 2a = 0$ (where $t_1t_2 = -1$) |
| Chord with Midpoint | Midpoint $(x_M, y_M)$ | $yy_M - 2a(x + x_M) = y_M^2 - 4ax_M$ |
Practical Insights
- Slope Determination: The slope of a chord connecting two parametric points $P_1(t_1)$ and $P_2(t_2)$ is $2/(t_1 + t_2)$. This relation is fundamental for problems involving the slope of chords.
- Perpendicular Focal Chords: If two focal chords are perpendicular, the product of their slopes is -1. This property can be used to derive relationships between their parameters.
- Geometric Applications: Chords are essential components in understanding various geometric properties of parabolas, including the formation of segments, properties related to tangents and normals, and their applications in fields like optics (parabolic reflectors) and structural engineering.
The equation of a chord of a parabola is not a single, fixed formula but rather a set of specialized equations tailored to the information available about the specific chord in question. The parametric and midpoint forms offer powerful tools for solving complex problems involving parabolas.
