Complex Numbers
Locus in complex plane
Grade 11
Question:
<p>If \(|(z - z_1)/(z - z_2)| = 3\), where \(z_1\) and \(z_2\) are fixed complex numbers and \(z\) is a variable complex number, then \(z\) lies on a</p>
<p>circle with \(z_1\) as its interior point</p>
<p>circle with \(z_2\) as its interior point</p>
<p>circle with \(z_1\) as its exterior point</p>
<p>circle with \(z_2\) as its exterior point</p>
Step-by-Step Solution
Key Concept: The locus of points satisfying |z - z₁|/|z - z₂| = k (constant) is a circle, derived from the Apollonius circle theorem: |z - z₁| = k|z - z₂| represents a circle whose radius and center depend on k, z₁, and z₂.
<p><strong>Step 1:</strong> Start with the given condition: |z - z₁|/|z - z₂| = 3</p><p><strong>Step 2:</strong> Rewrite as |z - z₁| = 3|z - z₂|. Square both sides: |z - z₁|² = 9|z - z₂|²</p><p><strong>Step 3:</strong> Let z = x + iy, z₁ = x₁ + iy₁, z₂ = x₂ + iy₂. Expanding: (x - x₁)² + (y - y₁)² = 9[(x - x₂)² + (y - y₂)²]</p><p><strong>Step 4:</strong> Rearrange and complete the square. This yields an equation of the form (x - h)² + (y - k)² = r², which is the standard circle equation.</p><p><strong>Step 5:</strong> Since the coefficient of the quadratic terms become equal and positive after simplification, the locus is definitely a <strong>circle</strong>.</p><p>∴ Answer: <strong>BD</strong> (Circle)</p>
Correct Answer: BD