Chapter 4 Exercise 18
- Katie Carlos
Suppose 2 circles intersect a two points P and Q. Prove the radical axis of these circles is PQ. Given 2 circles C1 and C2 that intersect at points P and Q. Power (P, C1)= d^2-r^2=r^2-r^2. Power (P,C1)=0. Power (P,C2)=d^2-r^2=r^2-r^2. Power (P,C2)=0. Since the Power (P,C1)= Power (P,C2). P lies on the radical axis for the circles C1 and C2. We also know that the Power (Q,C1)=d^2-r^2=r^2-r^2=0 and Power (Q,C2) =d^2-r^2=r^2-r^2=0. Thus the Power of(Q,C1)= Power (Q, C2). Hence, Q also lies on the radical axis for the circles C1 and C2. The radical axis is a line and these two points P and Q determine that line.
Let X be any point on the line PQ then an Alternative way to determine the Power (X, C1)= XP*XQ from Activity 5. The Power (X, C2)= XP*XQ. Since this is true for any arbitrary point X on the line PQ all points on this line belong the radical axis.