<p>Let \(z_1\) and \(z_2\) be two complex numbers satisfying \(|z_1| = 9\) and \(|z_2 - 3 - 4i| = 4\). Then the minimum value of \(|z_1 - z_2|\) is __________.</p>
Step-by-Step Solution
Key Concept: z₁ lies on a circle of radius 9 centered at origin, and z₂ lies on a circle of radius 4 centered at (3,4). The minimum distance between z₁ and z₂ equals the distance between circle centers minus the sum of radii.
<p><strong>Step 1:</strong> Interpret the constraints geometrically.</p><p>|z₁| = 9 means z₁ lies on a circle C₁ with center O = (0,0) and radius r₁ = 9.</p><p>|z₂ - 3 - 4i| = 4 means z₂ lies on a circle C₂ with center (3,4) and radius r₂ = 4.</p><p><strong>Step 2:</strong> Find the distance between the two circle centers.</p><p>Distance d = |3 + 4i - 0| = √(3² + 4²) = √(9 + 16) = √25 = 5.</p><p><strong>Step 3:</strong> Apply the geometric principle for minimum distance between points on two circles.</p><p>When two circles are external to each other (or one contains the other), the minimum distance between a point on C₁ and a point on C₂ is:</p><p>|z₁ - z₂|₍min₎ = d - r₁ - r₂ (if circles are external)</p><p>Here: d = 5, r₁ = 9, r₂ = 4</p><p>Since d = 5 < r₁ = 9, the center of C₂ lies inside C₁, so we use:</p><p>|z₁ - z₂|₍min₎ = r₁ - d - r₂ = 9 - 5 - 4 = 0</p><p><strong>Alternative:</strong> If 5 < 9 - 4 = 5 (boundary case), C₂ is internally tangent to C₁.</p><p>The minimum occurs when z₁, center of C₁, center of C₂, and z₂ are collinear with z₁ and z₂ on the same side:</p><p>|z₁ - z₂|₍min₎ = |r₁ - (d + r₂)| = |9 - (5 + 4)| = |9 - 9| = 0</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0