Complex Numbers
Geometry of complex numbers
Grade 11
Question:
<p>The largest value of \(r\) for which the region represented by the set \(\{\omega \in \mathbb{C} : |\omega - 4 - i| \leq r\}\) is contained in the region represented by the set \(\{z \in \mathbb{C} : |z - 1| \leq |z + i|\}\), is equal to</p>
<p>\(\sqrt{17}\)</p>
<p>\(2\sqrt{2}\)</p>
<p>\(\dfrac{3}{2}\sqrt{2}\)</p>
<p>\(\dfrac{5}{2}\sqrt{2}\)</p>
Step-by-Step Solution
Key Concept: The first set is a closed disk centered at 4+i with radius r. The second set is a half-plane defined by the perpendicular bisector of points 1 and -i. The largest r occurs when the disk is internally tangent to the boundary line of the half-plane.
<p><strong>Step 1:</strong> Identify the second region. The condition |z-1| ≤ |z+i| represents points closer to (1,0) than to (0,-1). Squaring both sides:</p><p>|z-1|² ≤ |z+i|²</p><p>(x-1)² + y² ≤ x² + (y+1)²</p><p>x² - 2x + 1 + y² ≤ x² + y² + 2y + 1</p><p>-2x ≤ 2y</p><p><strong>x + y ≥ 0</strong> (or x + y = 0 is the boundary line)</p><p><strong>Step 2:</strong> The first region is a disk with center C = (4, 1) and radius r. For this disk to be contained in the half-plane x + y ≥ 0, the center must be in the half-plane and r cannot exceed the distance from C to the boundary line x + y = 0.</p><p><strong>Step 3:</strong> Check center: 4 + 1 = 5 > 0 ✓ (center is in the half-plane)</p><p><strong>Step 4:</strong> Calculate distance from (4, 1) to line x + y = 0:</p><p>d = |4 + 1 - 0|/√(1² + 1²) = 5/√2 = (5√2)/2</p><p><strong>Step 5:</strong> The maximum radius is when the disk is tangent to the boundary line:</p><p>r_max = (5√2)/2</p><p>∴ Answer: D</p>
Correct Answer: D