Cambridge O Level Additional Mathematics · Syllabus 4037 · Coordinate Geometry of the Circle
Common Chord
What is Common Chord?
The line segment joining the two points at which two intersecting circles cross. Its equation is found by writing both circle equations in the expanded general form and subtracting one from the other: because the coefficients of x squared and y squared are the same in both, those terms cancel and a linear equation remains, which is the line through both points of intersection. Substituting that line back into either circle then gives the points themselves, and each point should be checked in both original equations. The subtraction can always be carried out, even when the circles do not meet, so the resulting line only deserves the name common chord when real intersection points actually exist; otherwise it is a line that cuts neither circle. When the circles do intersect, the line of centres is the perpendicular bisector of the common chord.
This definition is part of the Coordinate Geometry of the Circle chapter in Cambridge O Level Additional Mathematics.