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 yintercept of the chord.
3. Finding the Equation of Chords:
 Given two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ on the parabola, the equation of the chord passing through these points can be found using the formula for the slopeintercept 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 $(h,\frac{1}{4a}+k)$ must coincide with the focus of the parabola $(h,\frac{1}{4a})$.
 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 $(0,\frac{1}{4a})$.
 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.