Exercise 18.  Show that the parabola    is mapped onto the cardioid    by the reciprocal transformation.  

Solution 18.

See text and/or instructor's solution manual.

Solution.  The inverse mapping is  .   

Now use the substitution    and get:

Now use the quadratic equation and solve for    and get:

          

Therefore, the image of the parabola    under the mapping    is the cardioid  .

We are done.   

Aside.  We can let Mathematica double check our work.

                    Graph of the parabola  ,  using the parametric equations  .  

                    Graph of the image of the parabola  ,  using the parametric equations  ,
                    and   ,    and    ,  
                    and this graph can also be obtained using the polar coordinate form   .  

Aside.  We can let Mathematica double check our work.

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

This solution is complements of the authors.

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