Solution 2 (a).

See text and/or instructor's solution manual.

Use the relations   ,   and  

                        and    ,     

and polar coordinates    ,    ,    and       

and then get the equations    

                        and    .  

Solve these equations for   ,      and get     

                        and    ,    respectively.   

Use the trigonometric identity      to eliminate ,  

                    ,  

and  obtain the equation of an ellipse   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can graph the images of some circles under the mapping   .  

                    The image of the circles   ,   (for  ),   are mapped onto the ellipses   

                    the ellipses      by the Joukowski transformation   .   

This solution is complements of the authors.

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