Common Chord of Two Circles

Common Chord of Two Circles in Coordinate Geometry:

1. Equation of Circles:

  • Two circles with centers (h1,k1) and (h2,k2) and radii r1 and r2 respectively are represented by their equations:
    • Circle 1: (xh1)2+(yk1)2=r12
    • Circle 2: (xh2)2+(yk2)2=r22

2. Common Chord:

  • The common chord of two circles is a straight line segment that is a chord of both circles.

3. Finding the Common Chord:

  • Step 1: Circle Equations: Start by obtaining the equations of both circles.
  • Step 2: Simultaneous Equations: Solve the equations simultaneously to find the points of intersection, which represent the endpoints of the common chord.
  • Step 3: Determine the Chord Equation: Use the coordinates of the intersection points to find the equation of the common chord in the form y=mx+c.

4. Midpoint of Common Chord:

  • The midpoint of the common chord lies on the line joining the centers of the circles and is equidistant from both circles.

5. Length of Common Chord:

  • The distance between the points of intersection (endpoints of the common chord) can be calculated using the distance formula: d=(x2x1)2+(y2y1)2.

6. Perpendicular Bisector of the Common Chord:

  • The perpendicular bisector of the common chord passes through the midpoint of the chord and is perpendicular to it.