Complex Numbers
If z is a complex number such that |z| = 1, then the value of (1 + z)/(1 + z̄) (where z̄ is the conjugate of z) lies on
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Solution
The unit circle centered at origin
- Modulus Constraint: ∣z∣=1 implies z⋅zˉ=∣z∣2=1⟹zˉ=1/z.
- Substitution:
- 1+zˉ1+z=1+1/z1+z
- =(z+1)/z1+z=1+zz(1+z)
- Result: The expression simplifies exactly to z.
- Conclusion: Since the expression equals z, and ∣z∣=1, the value lies on the unit circle centered at the origin.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- complex numbers
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
The unit circle centered at origin
- Modulus Constraint: ∣z∣=1 implies z⋅zˉ=∣z∣2=1⟹zˉ=1/z.
- Substitution:
- 1+zˉ1+z=1+1/z1+z
- =(z+1)/z1+z=1+zz(1+z)
- Result: The expression simplifies exactly to z.
- Conclusion: Since the expression equals z, and ∣z∣=1, the value lies on the unit circle centered at the origin.
This hard difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of complex numbers. It appeared in the 2025 exam.
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