Equation of Chord with Midpoints

When studying conic sections such as parabolas, the equation of a chord with given midpoints is an important concept. A chord is a line segment with both endpoints on the curve, and the midpoint is the point that divides the chord into two equal parts.

Understanding the Equation of a Chord

For a parabola defined by the equation (y^2 = 4ax), the equation of any chord can be expressed in terms of its midpoint's coordinates ((h, k)). The derivation of the equation of the chord with midpoint ((h, k)) is based on the property that the midpoint of a chord lies on the line that is the geometric mean between the parabola and its tangent at the vertex, which is the axis of the parabola.

Derivation

Let (P(x_1, y_1)) and (Q(x_2, y_2)) be the endpoints of the chord, and let (M(h, k)) be the midpoint. By the midpoint formula, we have:

[ h = \frac{x_1 + x_2}{2} \quad \text{and} \quad k = \frac{y_1 + y_2}{2} ]

For the parabola (y^2 = 4ax), the endpoints (P) and (Q) satisfy the equation, so:

[ y_1^2 = 4ax_1 \quad \text{and} \quad y_2^2 = 4ax_2 ]

Adding these two equations, we get:

[ y_1^2 + y_2^2 = 4a(x_1 + x_2) ]

Using the midpoint coordinates, this simplifies to:

[ 2k^2 = 4a(2h) ]

[ k^2 = 4ah ]

This is the equation of the chord in terms of the midpoint coordinates ((h, k)).

Equation of the Chord with Midpoint ((h, k))

The equation of the chord with midpoint ((h, k)) for the parabola (y^2 = 4ax) is given by:

[ T = 0 ]

Where (T) is the expression obtained by replacing (x) with (x + h) and (y) with (y + k) in the equation of the parabola. So, for (y^2 = 4ax), we have:

[ (y + k)^2 = 4a(x + h) ]

Expanding and simplifying, we get:

[ yy_1 + yy_2 = 2a(x + x_1 + x_2) ]

Since (y_1 + y_2 = 2k) and (x_1 + x_2 = 2h), we can write:

[ y(k) = a(x + h) ]

This is the final equation of the chord with midpoint ((h, k)).

Table of Differences and Important Points

Aspect	General Line	Chord with Midpoint ((h, k))
Definition	A straight line with no fixed relationship to a curve.	A line segment with both endpoints on the parabola.
Equation	(y = mx + c) where (m) is the slope and (c) is the y-intercept.	(y(k) = a(x + h)) where ((h, k)) is the midpoint of the chord.
Relation to Parabola	May or may not intersect the parabola.	Always intersects the parabola at two points.
Midpoint Coordinates	Not necessarily related to the line equation.	Directly used to derive the equation of the chord.
Dependence on Parabola	Independent of the parabola's equation.	Dependent on the parabola's equation and properties.

Examples

Example 1: Find the Equation of the Chord

Given a parabola (y^2 = 12x) and a midpoint of the chord ((2, 6)), find the equation of the chord.

Solution:

Using the formula (y(k) = a(x + h)), we substitute (a = 3) (from (y^2 = 12x)), (h = 2), and (k = 6) to get:

[ y(6) = 3(x + 2) ]

[ 6y = 3x + 6 ]

[ y = \frac{1}{2}x + 1 ]

This is the equation of the chord with midpoint ((2, 6)).

Example 2: Verify the Midpoint

Given the parabola (y^2 = 8x) and the chord with equation (4y = x + 4), verify that the midpoint of the chord is ((4, 2)).

Solution:

First, we express the chord equation in the form (y(k) = a(x + h)):

[ 4y = x + 4 ]

[ y = \frac{1}{4}x + 1 ]

Now, compare this with (y(k) = a(x + h)), where (a = 2) (from (y^2 = 8x)). We see that (k = 1) and (h = 4).

To verify the midpoint, we plug (h = 4) and (k = 1) into the parabola's equation:

[ k^2 = 4ah ]

[ 1^2 = 4(2)(4) ]

[ 1 = 32 ]

This does not hold true, so there was a mistake in our comparison. The correct values should be (k = 2) and (h = 4). Let's verify:

[ k^2 = 4ah ]

[ 2^2 = 4(2)(4) ]

[ 4 = 32 ]

This is still incorrect, which means the given midpoint ((4, 2)) does not correspond to the chord (4y = x + 4) for the parabola (y^2 = 8x). The correct midpoint can be found by solving the system of equations given by the parabola and the chord.

These examples illustrate how to find the equation of a chord with a given midpoint and how to verify the midpoint of a given chord for a specific parabola. Understanding these concepts is essential for solving problems related to chords and parabolas in exams.