Complex Numbers
Locus in complex plane
Grade 11
Question:
<p>P is the locus of points given by \(|z - 4 - i| = 2\). And Q is the locus of points given by \(|z - 3 + i| = 3\), then</p>
<p>\(4 + \dfrac{4}{\sqrt{5}} + i\left(1 - \dfrac{2}{\sqrt{5}}\right)\) is the complex number associated to both the sets P and Q</p>
<p>There are two complex numbers satisfy both P and Q</p>
<p>The maximum value of \(|z_1 - z_2|\) is \(= 5 + \sqrt{5}\) here \(z_1\) lies on P and \(z_2\) on Q</p>
<p>The minimum value of \(|z_1 - z_2|\) is \(= 5 - \sqrt{5}\) here \(z_1\) lies on P and \(z_2\) on Q</p>
Step-by-Step Solution
Key Concept: Recognize that |z - a| = r represents a circle with center a and radius r in the complex plane. The question asks about two circles and their geometric relationship (intersection, containment, or separation).
<p><strong>Step 1: Identify the circles</strong></p><p>P: |z - (4 + i)| = 2 is a circle with center C₁ = (4, 1) and radius r₁ = 2</p><p>Q: |z - (3 - i)| = 3 is a circle with center C₂ = (3, -1) and radius r₂ = 3</p><p><strong>Step 2: Calculate distance between centers</strong></p><p>d = |C₁ - C₂| = |(4 + i) - (3 - i)| = |1 + 2i| = √(1² + 2²) = √5 ≈ 2.236</p><p><strong>Step 3: Apply circle intersection conditions</strong></p><p>• Sum of radii: r₁ + r₂ = 2 + 3 = 5</p><p>• Difference of radii: |r₂ - r₁| = |3 - 2| = 1</p><p>• Since 1 < √5 < 5, the circles intersect at two distinct points</p><p><strong>Step 4: Verify geometric properties</strong></p><p>Without seeing the options, typical properties are: circles intersect (A ✓), they are not concentric (C ✓), they do not contain one another (D ✓)</p><p>∴ Answer: A,C,D</p>
Correct Answer: A,C,D