Example ① For a point z on |z| = 1, find the range traced by w = z + 1/z.
Solution:If z=cosθ+i sinθ, then 1/z=cosθ−i sinθ, so w=2cosθ。
Therefore, w traces the line segment on the real axis satisfying −2 ≦ w ≦ 2。
Solution:If z=cosθ+i sinθ, then 1/z=cosθ−i sinθ, so w=2cosθ。
Therefore, w traces the line segment on the real axis satisfying −2 ≦ w ≦ 2。
Key Point
- When |z|=1, 1/z = z̄
Practice Problems
- Describe the locus represented by z satisfying |z−1| = 2|z+1|。
Answers and Explanations
- Answer:An Apollonius circle. Squaring both sides and simplifying gives a circle with center (−5/3, 0) and radius 4/3.