The Art and Craft of Problem Solving

128 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICS

Solution: Our solution illustrates a useful application of viewing complex num­

bers as vectors for parametrizing curves in the plane. Consider the general case, illus­

trated in the figure below, of two circles situated in the complex plane, with centers at

a, b and radii u , v, respectively. Notice that a and b are complex numbers, while u and

v are real. We are assuming, as in the original problem, that v is quite a bit bigger than

u.

The locus we seek is the set of midpoints M of the line segments XY, where X can be

any point on the left circle and Y can be any point on the right circle.

Thus, X = a + ueit and Y = b + veis, where t and s can be any values between 0

and 2n. We have

X +Y a+b ueit +veis

M=

• 2
  
  
  • = -
    2
    
    
    • 
      
      
      
      • 2
        Let us interpret this geometrically, by first trying to understand what
        ueit +veis
      
      
      
      
    
    
    
    
  
  
  
  

looks like as 0 :S s, t :S 2n. If we fix s, then P = veis is a point on the circle with radius

v centered at the origin. Now, when we add ueit to P, and let t vary between 0 and 2n,

we will get a circle with radius u, centered at P (shown as a dotted line below).

o -----'---..---1

Now let s vary as well. The small dotted circle will travel all along the circumference

of the large circle, creating an annulus or filled-in "ring." In other words, the locus of

points