Question

For α, β, z ∈ C and λ > 1, if \(\sqrt{λ - 1}\) is the radius of the circle |z - α|² + |z - β|² = 2λ, then |α - β| is equal to _____.

Answer 1

Correct Answer: 2

We are given the equation of a circle in the complex plane:

\[ |z - \alpha|^2 + |z - \beta|^2 = 2\lambda. \]

This represents a circle with center at the midpoint of \( \alpha \) and \( \beta \), and radius \( \sqrt{\lambda - 1} \).

The standard form of such a circle is:

\[ R^2 = \frac{|\alpha - \beta|^2}{4} + (\lambda - 1). \]

Since the given radius is \( \sqrt{\lambda - 1} \), we equate:

\[ \lambda - 1 = \frac{|\alpha - \beta|^2}{4} + (\lambda - 1). \]

Canceling \( \lambda - 1 \) on both sides:

\[ \frac{|\alpha - \beta|^2}{4} = 1. \]

Solving for \( |\alpha - \beta| \):

\[ |\alpha - \beta| = 2. \]

Final Answer: \( |\alpha - \beta| = 2 \).