prove that inversion about the circle given by x2y2 1 takes

prove that inversion about the circle given by x^2+y^2 = 1 takes the point (x,y) does not equal 0 into the point x/(x^2+y^2), y/(x^2+y^2).

Solution

An inversion in a circle is referred to simply as an inversion. Notice that points inside the circle and close to the center, O, get sent to points that are very far away from the circle. In addition, it can be easily verified that points on C are fixed by inversion in C. However, there is one point in the plane that does not have an image under inversion, O. As P gets closer to O, P 0 get farther away from O, so in some sense, we can think of O as mapping to a point at infinity. we say that the center of the circle of inversion, O, is mapped to a “point at infinity” on the extended plane, and O will be called the center of the inversion.