# Chords of a Parabola

### Chords of a Parabola:

1. Definition:

• A chord of a parabola is a straight line segment that connects two points on the parabolic curve.
• It intersects the parabola at two distinct points and lies entirely within the curve.

2. Equation of a Chord:

• The general equation of a chord of a parabola, y = ax² + bx + c, is expressed as: $y=mx+k$
• where $m$ and $k$ are constants representing the slope and y-intercept of the chord.

3. Finding the Equation of Chords:

• Given two points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ on the parabola, the equation of the chord passing through these points can be found using the formula for the slope-intercept form of a line.

4. Condition for a Chord to Be a Focal Chord:

• A focal chord of a parabola is a chord that passes through the focus of the parabola.
• Condition for a Chord to Be a Focal Chord:
• If a chord with equation $y=mx+k$ is a focal chord of the parabola $y=a{x}^{2}+bx+c$:
• The midpoint of the chord $\left(h,\frac{1}{4a}+k\right)$ must coincide with the focus of the parabola $\left(h,\frac{1}{4a}\right)$.
• Therefore, $k=\frac{1}{4a}-\frac{c}{2}$ for the chord to be a focal chord.

4. Deriving Conditions for Focal Chords:

Given a Parabola Equation: $y=a{x}^{2}$

• The focus is at $\left(0,\frac{1}{4a}\right)$.
• For a chord to be a focal chord, its midpoint should coincide with the focus.

5. Characteristics:

• A focal chord passes through the focus of the parabola and is perpendicular to the axis of symmetry.
• Its midpoint coincides with the focus, and it divides the parabola symmetrically.

6. Graphical Representation:

• On a graph, draw the parabola and then plot different chords connecting various points on the curve.
• Identify focal chords by verifying if their midpoints align with the focus of the parabola.