Complex Numbers
Unimodular Complex Numbers
Grade 11
Question:
<p>A complex number \(z\) is said to be unimodular if \(|z| = 1\). Suppose \(z_1\) and \(z_2\) are complex numbers such that \(\dfrac{z_1 - 2z_2}{2 - z_1\bar{z}_2}\) is unimodular and \(z_2\) is non-unimodular. Then the point \(z_1\) lies on a</p>
<p>straight line parallel to \(x\)-axis</p>
<p>straight line parallel to \(y\)-axis</p>
<p>circle of radius \(2\)</p>
<p>circle of radius \(\sqrt{2}\)</p>
Step-by-Step Solution
Key Concept: If a complex number w is unimodular (|w| = 1), then w·w̄ = 1. Apply this to the given fraction and use the algebraic identity |a|² = a·ā to derive a geometric condition on z₁.
<p><strong>Step 1:</strong> Let w = (z₁ - 2z₂)/(2 - z₁z̄₂). Since w is unimodular, |w| = 1, which means w·w̄ = 1.</p><p><strong>Step 2:</strong> We have w̄ = (z̄₁ - 2z̄₂)/(2 - z̄₁z₂). Computing w·w̄ = 1:</p><p>(z₁ - 2z₂)(z̄₁ - 2z̄₂) = (2 - z₁z̄₂)(2 - z̄₁z₂)</p><p><strong>Step 3:</strong> Expand LHS: |z₁|² - 2z₁z̄₂ - 2z̄₁z₂ + 4|z₂|²</p><p>Expand RHS: 4 - 2z̄₁z₂ - 2z₁z̄₂ + |z₁|²|z₂|²</p><p><strong>Step 4:</strong> Simplify: |z₁|² + 4|z₂|² = 4 + |z₁|²|z₂|²</p><p>Rearrange: |z₁|² - |z₁|²|z₂|² = 4 - 4|z₂|²</p><p>Factor: |z₁|²(1 - |z₂|²) = 4(1 - |z₂|²)</p><p><strong>Step 5:</strong> Since z₂ is non-unimodular, |z₂|² ≠ 1, so (1 - |z₂|²) ≠ 0.</p><p>Divide both sides: |z₁|² = 4</p><p>Therefore: |z₁| = 2</p><p>∴ z₁ lies on a <strong>circle with center at origin and radius 2</strong></p>
Correct Answer: C