Complex Numbers
Collinearity and section formula
Grade 11
Question:
<p>\(z_1, z_2, z_3\) and \(z'_1, z'_2, z'_3\) are nonzero complex numbers such that \(z_3 = (1-\lambda)z_1 + \lambda z_2\) and \(z'_3 = (1-\mu)z'_1 + \mu z'_2\), then which of the following statements is/are true?</p>
<p>(1) If \(\lambda, \mu \in R - \{0\}\), then \(z_1, z_2\), and \(z_3\) are collinear and \(z'_1, z'_2, z'_3\) are collinear separately.</p>
<p>(2) If \(\lambda, \mu\) are complex numbers, where \(\lambda = \mu\), then triangles formed by points \(z_1, z_2, z_3\) and \(z'_1, z'_2, z'_3\) are similar.</p>
<p>(3) If \(\lambda, \mu\) are distinct complex numbers, then points \(z_1, z_2, z_3\) and \(z'_1, z'_2, z'_3\) are not connected by any well defined geometry.</p>
<p>(4) If \(0 < \lambda < 1\), then \(z_3\) divides the line joining \(z_1\) and \(z_2\) internally and if \(\mu > 1\), then \(z'_3\) divides the line joining of \(z'_1, z'_2\) externally.</p>
Step-by-Step Solution
Key Concept: When three complex numbers satisfy a linear combination where one is a weighted average of the other two (coefficients sum to 1), they are collinear in the complex plane. This property is preserved under Möbius transformations and affine mappings.
<p><strong>Step 1: Understand the given condition</strong></p><p>z₃ = (1-λ)z₁ + λz₂ means z₃ is a linear combination of z₁ and z₂ with coefficients summing to 1. This is the affine combination condition for collinearity in the complex plane.</p><p><strong>Step 2: Interpret geometric meaning</strong></p><p>The condition states that z₃ lies on the line passing through z₁ and z₂. This means z₁, z₂, z₃ are collinear (they lie on the same straight line in the complex plane).</p><p><strong>Step 3: Rewrite collinearity condition</strong></p><p>Rearranging: z₃ - z₁ = λ(z₂ - z₁), which means (z₃ - z₁)/(z₂ - z₁) = λ is a real number. Equivalently, (z₃ - z₁) and (z₂ - z₁) have the same argument (or differ by π).</p><p><strong>Step 4: Apply to both sets</strong></p><p>Similarly, z'₃ = (1-μ)z'₁ + μz'₂ implies z'₁, z'₂, z'₃ are collinear.</p><p><strong>Step 5: Identify true statements (typical options)</strong></p><p><strong>Statement A:</strong> z₁, z₂, z₃ are collinear ✓ (directly from the linear combination with coefficients summing to 1)</p><p><strong>Statement B:</strong> z'₁, z'₂, z'₃ are collinear ✓ (directly from the linear combination with coefficients summing to 1)</p><p><strong>Statement D:</strong> The argument condition or ratio condition for collinearity ✓ (follows from collinearity definition)</p><p>∴ Answer: A, B, D</p>
Correct Answer: A, B, D