<p>If \(\left|\dfrac{z_1 - 2z_2}{2 - z_1\bar{z}_2}\right| = 1\) and \(|z_2| \neq 1\), then \(|z_1|\) equals:</p>
Step-by-Step Solution
Key Concept: This expression has the form of a Möbius transformation. When |w| = 1 for w = (z₁ - 2z₂)/(2 - z₁z̄₂), we can use the property that |w| = 1 implies w·w̄ = 1, and apply the conjugate relationship to find |z₁|.
<p><strong>Step 1:</strong> Given: $\left|\dfrac{z_1 - 2z_2}{2 - z_1\bar{z}_2}\right| = 1$ with $|z_2| \neq 1$</p><p>This means: $|z_1 - 2z_2| = |2 - z_1\bar{z}_2|$</p><p><strong>Step 2:</strong> Square both sides:</p><p>$|z_1 - 2z_2|^2 = |2 - z_1\bar{z}_2|^2$</p><p><strong>Step 3:</strong> Expand the left side:</p><p>$(z_1 - 2z_2)(\bar{z}_1 - 2\bar{z}_2) = |z_1|^2 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + 4|z_2|^2$</p><p><strong>Step 4:</strong> Expand the right side:</p><p>$(2 - z_1\bar{z}_2)(2 - \bar{z}_1z_2) = 4 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + |z_1|^2|z_2|^2$</p><p><strong>Step 5:</strong> Set them equal:</p><p>$|z_1|^2 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + 4|z_2|^2 = 4 - 2z_1\bar{z}_2 - 2\bar{z}_1z_2 + |z_1|^2|z_2|^2$</p><p><strong>Step 6:</strong> The middle terms cancel. Simplify:</p><p>$|z_1|^2 + 4|z_2|^2 = 4 + |z_1|^2|z_2|^2$</p><p><strong>Step 7:</strong> Rearrange:</p><p>$|z_1|^2 - |z_1|^2|z_2|^2 = 4 - 4|z_2|^2$</p><p>$|z_1|^2(1 - |z_2|^2) = 4(1 - |z_2|^2)$</p><p><strong>Step 8:</strong> Since $|z_2| \neq 1$, we have $(1 - |z_2|^2) \neq 0$. Divide both sides:</p><p>$|z_1|^2 = 4$</p><p>$|z_1| = 2$</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B