Exercise 3.  Find the image of the vertical line    under the transformation   .  

Solution 3.

Answer.   The image of      under      is the right branch of the hyperbola  .  

Hint.   Use ideas found in Example 10.13.  

Short Solution.   Substitute      into equation (10-23)     

and get the hyperbola       which reduces to    .

We might be done.   

Solution.   Using Equation (5-33), we write  

                    .  

The image of the vertical line    is the curve in the w plane given by the parametric equations  

                    ,    for  .  

Next, we rewrite these equations as  

                    .  

Eliminate y from these equations by squaring and using the hyperbolic identity  .  

The result is the single equation    .   

Now use      and get   .  

        Therefore, the image of the vertical line      under the mapping      

is the right branch of the hyperbola   .  

We are done.   

Aside.  We can look at some graphs of the mapping  .

                      The image of    under    is the right branch of the hyperbola  .  

                    The image of    under    is the right branch of the hyperbola  .  

This solution is complements of the authors.

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