Problem 1 · Compare squares to reveal a complex number’s real part · 3 points
Made by AI and not reviewed by a real person.
A complex number z has |z| = 2, and (z + 1)² and (z − 1)² have the same real part. What is (z + i)² + (z − i)²?
- −10
- −6
- 6
- 10
- −8
Show a hint
- What does the matching real part tell you about the difference of the two squares involving 1?
- Use that difference and |z| to find z² before expanding the expression you need.
Show the answer and solution
Answer: A — −10
The difference (z + 1)² − (z − 1)² equals 4z. Since the two squares have the same real part, z has real part 0. With |z| = 2, this gives z = ±2i and z² = −4. Thus (z + i)² + (z − i)² = 2z² + 2i² = −8 − 2 = −10. The tempting −6 results from using i² = 1 in the last step.