Complex Numbers
Modulus of complex numbers
Grade 11
Question:
<p>Let four points \(z_1, z_2, z_3, z_4\) be in complex plane such that \(|z_1| < 1\), \(|z_2| = 1\) and \(|z_3| \leq 1\). If \(z_3 = \dfrac{z_2(z_1 - z_4)}{\bar{z}_1 z_4 - 1}\), then \(|z_4|\) can be</p>
<p>2</p>
<p>\(\dfrac{2}{5}\)</p>
<p>\(\dfrac{1}{3}\)</p>
<p>\(\dfrac{5}{2}\)</p>
Step-by-Step Solution
Key Concept: Use the geometric property that equal moduli represent points on circles centered at origin, and equal arguments represent points on rays from origin. The constraint |z₁| < |z₂| < |z₃| < |z₄| with arg(z₁) = arg(z₂) = arg(z₃) = arg(z₄) means all four points are collinear on the same ray from origin, with z₁ closest and z₄ farthest.
<p><strong>Step 1:</strong> Interpret the conditions. We have |z₁| < |z₂| < |z₃| < |z₄| (strictly increasing moduli) and arg(z₁) = arg(z₂) = arg(z₃) = arg(z₄) = θ (same argument).</p><p><strong>Step 2:</strong> Equal arguments mean all four complex numbers point in the same direction from the origin. Each zₖ can be written as zₖ = rₖe^(iθ) where r₁ < r₂ < r₃ < r₄.</p><p><strong>Step 3:</strong> Geometrically, z₁, z₂, z₃, z₄ all lie on the same ray emanating from the origin at angle θ, with z₁ closest to origin and z₄ farthest. They are collinear points on this ray.</p><p><strong>Step 4:</strong> This arrangement represents four points on a single ray from the origin, ordered by distance. Any property asking about their configuration must account for this collinear arrangement on the same half-line.</p><p>∴ Answer: D</p>
Correct Answer: D