Intersection Points of Two Circles

Write both equations with coefficients 1 on x2,y2, then subtract. For distinct centres this gives a line through any common points. (D_1-D_2)x+(E_1-E_2)y+(F_1-F_2)=0 Substitute the line into either circle equation. Solve the quadratic, then find both coordinates of each intersection. Two, one or no real roots correspond to two intersections, tangency or no intersection. For concentric circles, compare their radii directly. If the circles cross at P and Q, subtraction gives the equation of the common chord line PQ. Its segment ends at the two intersections; substituting into either circle finds them. The line of centres is perpendicular to this chord and bisects it. For x2+y2=25 and (x-6)2+y2=25, subtracting gives 12x-36=0. Thus x=3, then y2=16. The common chord joins (3,-4) to (3,4) and has length 8.