If α > 1 > |w1| and |w1|² + |w2|² + α² = |w1|² + α²|w2|², where w1 and w2 are complex numbers and α is a real number, then

Steps to Solve

1. Understand the Given Equation

We have the equation: $$ |w_1|^2 |w_2|^2 + \alpha^2 = |w_1|^2 + \alpha^2 |w_2|^2 $$

2. Rearrange the Equation

Let's rearrange this equation to isolate terms related to the moduli. We subtract $|w_1|^2$ from both sides: $$ |w_1|^2 |w_2|^2 - |w_1|^2 + \alpha^2 - \alpha^2 |w_2|^2 = 0 $$

3. Factor the Expression

Now, we can factor the first two terms: $$ |w_1|^2 (|w_2|^2 - 1) + \alpha^2 (1 - |w_2|^2) = 0 $$

4. Set the Expression to Zero

For the equation to hold true, the two terms must sum to zero. This leads us to two possible cases:

• Case 1: $|w_2|^2 = 1$
• Case 2: $|w_1|^2 = 0$

5. Analyze the Cases

If $|w_2|^2 = 1$, then $|w_2| = 1$. If $|w_1|^2 = 0$, it means $|w_1| = 0$.

Given that $\alpha > 1 > |w_1|$, we find that:

• $|w_2|$ must relate to $\alpha$ while satisfying $|w_2| \leq \alpha$.

6. Check the Options

Now, we check the provided answer choices against the findings:

• Option A: $|w_1| = 1$ (incorrect because $|w_1| < 1$)
• Option B: $|w_2| = \alpha$ (could be true)
• Option C: $|w_1| = 1 + \alpha$ (incorrect, as $|w_1| < 1$)
• Option D: $|w_2| = 1$ (could be true but conflicts with given conditions on $\alpha$)