Exercise 7.  Find the image of the circle     under the reciprocal transformation  .  

Make sketches and indicate the points   in the z-plane and their images in the w-plane.

Solution 7.

See text and/or instructor's solution manual.

Answer.  The circle  .  

The points     are mapped onto  .

Graph of the circle  ,  using the parametric equations  ,  and the image circle  ,  using the parametric equations  ,  and  .  

Solution using algebra.

      The inverse mapping is  .   

Now use the substitution    and get:

This is an equation of the circle  .  

We are done.   

Remark.  The above solution depended on factoring the multivariate polynomial  .  
This is not as difficult as one might suspect because it must have as a factor.  

                    The circle    and the image circle  ,  divide the z-plane and w-plane into two pieces.

Solution using point images.  

      The points   lie on the circle    in the z-plane.  

Their image points are  ,  and these three points    determine the circle  

  in the w-plane.  

If we observe that the center of the circle    is  ,  then an easy calculation shows that

,  ,  and  ,  

and we have verified this observation.  

Aside.  We can let Mathematica double check our work.

                    The circle    and the image circle  .

                    The points     and  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell