Chord of a Circle

Chord of a Circle in Coordinate Geometry:

1. Circle Equation:

  • A circle with center (h,k) and radius r is represented by the equation (xh)2+(yk)2=r2.

2. Chord Equation:

  • A chord in a circle can be defined by the coordinates of its endpoints (x1,y1) and (x2,y2) lying on the circle.

3. Chord Length:

  • The distance between two points (x1,y1) and (x2,y2) can be found using the distance formula: d=(x2x1)2+(y2y1)2
  • . This represents the length of the chord.

4. Midpoint of a Chord:

  • The midpoint of the chord is the average of the x-coordinates and y-coordinates of the endpoints: 
  • To find the midpoint (xmid,ymid) of a chord with endpoints (x1,y1) and (x2,y2), use the midpoint formula:
    xmid=x1+x22andymid=y1+y22

5. Perpendicular Bisector of a Chord:

  • The line passing through the midpoint of a chord and perpendicular to the chord is the perpendicular bisector of the chord.
  • Its equation can be found using the negative reciprocal of the slope of the chord.

6. Intersecting Chords Theorem in Coordinates:

  • If two chords intersect within a circle, the product of the segments of one chord is equal to the product of the segments of the other. (AC × CB = EC × CD)

7. Slope of Chords:

  • The slope of the chord is calculated using the formula: m=y2y1x2x1.