If and are different complex numbers with , then find .
1
Explanation
This question tests the properties of modulus and conjugate of complex numbers. The key insight is to compute the squares of both the numerator and denominator moduli separately and show they are equal. Using the given condition |β| = 1 and properties like z·z̄ = |z|² and z̄₁z₂ = z̄₁·z̄₂ from the textbook, we can establish that |β - α|² = |1 - ᾱβ|², leading to the answer 1.
Solution Steps
Step 1: Let the given expression be E = |(β - α)/(1 - ᾱβ)|. Using property (ii) from the context: |z₁/z₂| = |z₁|/|z₂|, we get E = |β - α|/|1 - ᾱβ|.
Step 2: Calculate |β - α|² using the property z·z̄ = |z|²: |β - α|² = (β - α)(β̄ - ᾱ) = ββ̄ - βᾱ - αβ̄ + αᾱ = |β|² - βᾱ - αβ̄ + |α|². Since |β| = 1, we get |β - α|² = 1 - βᾱ - αβ̄ + |α|².
Step 3: Calculate |1 - ᾱβ|² similarly: |1 - ᾱβ|² = (1 - ᾱβ)(1 - ᾱβ̄) = (1 - ᾱβ)(1 - αβ̄) using property (iii): z̄₁z₂ = z̄₁·z̄₂. Expanding: = 1 - αβ̄ - ᾱβ + ᾱβ·αβ̄ = 1 - αβ̄ - ᾱβ + |α|²|β|².
Step 4: Since |β| = 1, we have |β|² = 1. Therefore, |1 - ᾱβ|² = 1 - αβ̄ - ᾱβ + |α|². Note that ᾱβ = βᾱ (commutative multiplication).
Step 5: Comparing results from Step 2 and Step 4: |β - α|² = |1 - ᾱβ|². Taking square root: |β - α| = |1 - ᾱβ|. Therefore, E = |β - α|/|1 - ᾱβ| = 1.