GEOMETRIA From the graph we can see that the two circles intersect in two distinguished point. The algebraic method. Let us solve the system with the equations of the two circles: x2 + y2 6x 6y + 8 = 0 {x2 + y2 2x 4y 8 = 0 The system is of fourth degree. We observe that the second-degree terms equal in the two equations thus we can apply the reduction method: x2 + y2 6x 6y + 8 = 0 {x2 + y2 2x 4y 8 = 0 4x 2y + 16 = 0 We have obtained a straight line that is called radical axis: it is the perpendicular line to the segment that has its end points in the centers of the circles passing through the intersection points. If we solve the system between the equation of the radical axis and one of the two circles equations, we will determine the coordinates of the points of intersections we are looking for. x2 + y2 6x 6y + 8 = 0 { 4x 2y + 16 = 0 y A 3 2 Again, by using the substitution method, we explicit y in the second equation and substitute the expression obtained in the first equation: x2 + y2 6x 6y + 8 = 0 {y = 2x + 8 We have to solve the following equation to find the system solutions: 6 5x2 26x + 24 = 0 x1 = __ e x2 = 4 5 If we substitute these values in the second equation, we will find the coordinates of the intersection points 6 28 of the two circles: A(__ ; ___) e B(4 ; 0). 5 5 Now, let us prove that the straight-line y = 2x + 8 (radical axis) is perpendicular to the line crossing the centers C1C2. Its angular coefficient is m = 2 while the one of the straight line crossing the two centers is y 1 y 2 ________ 3 2 _1_ = ; = m = _________ x1 x2 3 1 2 as their product is a 1, the two lines are perpendicular. C2 C1 B O 1 3 4 x x2 + ( 2x + 8)2 6x 6( 2x + 8) + 8 = 0 {y = 2x + 8 Exsercises. Determine, geometrically and algebraically, the relative positions of the circles of given equations and the equation of the radical axis. a. x2 + y2 4x + 2y + 4 = 0 e x2 + y2 + 6x + 4y + 4 = 0 b. x2 + y2 4x + 6y + 8 = 0 e x2 + y2 4x + 6y 5 = 0 c. Prove that, in general, given two circles of equations x2 + y2 + a1x + b1y + c1 = 0 and x2 + y2 + a2x + b2y + c2 = 0, their radical axis has equation (a1 a2)x + (b1 b2)y + (c1 c2) = 0 390