22 CHAPTER l • COMPLEX NUMBERS- Let z 1 and z 2 be two distinct points in the complex plane, and let K be a positive
real constant that is greater than the distance between z 1 and z 2.
(a) Show that the set of points {z: lz -zd + lz -z2I = K} is an ellipse
with foci z 1 and z2.
(b) Find the equation of the ellipse with foci ±3i t hat goes through the
point 8 - 3i.
(c) Find the equation of the ellipse with foci ±2i that goes through the
point 3 + 2i.- Supply the reason for the indicatt.'CI step in the proof of Theorem 1. 2.
1.4 The Geometry of Complex Numbers, Continued
In Section 1.3 we saw that a complex number z = x + iy could be viewed as a
vector in the xy plane with its tail at the origin and its head at the point (x, y).
A vector can be uniquely specified by giving its magnitude (i.e., its length) and
direction (i.e., the angle it makes with the positive x-axis). In this section, we
focus on these two geometric aspects of complex numbers.
Let r be the modulus of z (i.e., r = lzl), and let () be the angle tha t the line
from the origin to the complex number z makes with the positive x-axis. (Note:The number (J is undefined if z = 0.) Then, as Figure l.ll(a) shows,
z = (rcos9, rsin9) = r(cos(J + isin9). (1-27)
I Definition 1.9: Polar representation
Identity (1-27) is known as a polar representation of z, and the values r and
(J are called polar coordinates of z.y y(0,y)z = (x, y) = x + iy
- -"T =(r cos II, r sin II)= r(cos Ii+ i sin II)
z = (x, y) = x + iy
(0,y) • ·•• ;=(rcosll, r sinll) = r(cosll+ isin 9)(a) (b)
Figure 1.11 Polar representation of complex numbers.