Power of a point using Coordinates

(a) Let c1 be the circle with center O1(0,0) and radius r1. What is the power of c1 at P(x,y)? (i.e., give a formula for the power in terms of x and y)

(b) Let c2 be the circle with center O2(s,0) and radius r2. What is the power of c2 at P(x,y)? Note that this formula still is valid if r2=0, as in GTC, Chapter 9.

(c) Find the equation in x and y for the radical axis of c1 and c2.

Study Questions:  Know the answers and how to do the constructions

• What is the definition of the radical axis of two circles? (Note: the word "line" does not appear in this.)
• How do we know that the radical axis is a line?
• If two circles intersect, why does the radical axis pass through the points of intersection?
• Suppose that two circles c1 and c2 have a common tangent ST; then show the radical axis of the two circles passes through the midpoint of ST.
• In GTC, the radical axis of a circle C1 and a point P is defined. Explain why this is the usual definition in terms of powers of a point if you allow a "circle" to have radius 0.
• Let m be the radical axis of circle c1 and point P. If O is on m, explain why circle d, with center O and orthogonal to c1, must pass through P.
• Know this construction: Given a circle c1 and a point P1, construct a circle centered at P1 that is orthogonal to c1. (Is there more than one circle? Does it always exist?)
• Know this construction: Given a circle c1 and a point P1, construct a circle through P1 that is orthogonal to c1. (Is there more than one circle? Does it always exist? Notice that this construction is quite different from the previous one.)
• Figure out this set of related constructions. Given p circles and q points, find a circle (or circles) through the q points that is (are) orthogonal to each of the p circles. This works if p+q = 3 or less.
• Given two circles c1 and c2 and a point P, when can you construct a circle with center P that is orthogonal to both the given circles?