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 (x1,y1) and (x2,y2) 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=ax2+bx+c:
    • The midpoint of the chord (h,14a+k) must coincide with the focus of the parabola (h,14a).
    • Therefore, k=14ac2 for the chord to be a focal chord.

4. Deriving Conditions for Focal Chords:

Given a Parabola Equation: y=ax2

    • The focus is at (0,14a).
    • 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.