Chords of a Parabola
Chords of a Parabola:
- 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:
- where and are constants representing the slope and y-intercept of the chord.
3. Finding the Equation of Chords:
- Given two points and 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 is a focal chord of the parabola :
- The midpoint of the chord must coincide with the focus of the parabola .
- Therefore, for the chord to be a focal chord.
4. Deriving Conditions for Focal Chords:
Given a Parabola Equation:
- The focus is at .
- For a chord to be a focal chord, its midpoint should coincide with the focus.
- 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.