Exercise 25.  Let   and   be two distinct points in the complex plane, and let  K  be a positive real constant that is greater than the distance between   and   .

25 (c).  Find the equation of the ellipse with foci    that goes through the point  .

Solution 25 (c).

See text and/or instructor's solution manual.

Letting  ,   and  ,  we compute  .  

Then, with  ,  the equation in Exercise 25 (a)  becomes  

          

which can be written as    from which we obtain  .  

Show the details that squaring both sides, simplifying, squaring again, and simplifying again gives  .

In standard form,  .

Thus, we have obtained the formula    for the ellipse.  

Now a few more manipulations are used.

Hence, the standard form for the ellipse is   .

This solution is complements of the authors.

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