Solution 2 (b).

See text and/or instructor's solution manual.

Answer.   

Use the relations   ,  

and        and    .   

Set    and use polar coordinates    ,    and    ,  

get the equations   

                        and    ,

and write them in the form   

                        and    ,   respectively.  

Then square these terms and obtain  

                    .  

It is now easy to obtain the equation of a hyperbola   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

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

                    The image of the rays   ,   (for  ),   are mapped onto    

                    a branch of the hyperbola      by the Joukowski transformation   .   

We are really really done.   

                    Remark.  The images of      will be the same family of hyperbolas.

                    The image of the rays   ,   (for  ),   are mapped onto    

                    a branch of the hyperbola      by the Joukowski transformation   .   

This solution is complements of the authors.

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