<p>Find the minimum value of \(|z - 1|\) if \(\||z - 3| - |z + 1|\| = 2\).</p>
Step-by-Step Solution
Key Concept: The constraint ||z - 3| - |z + 1|| = 2 describes a hyperbola (the locus of points where the absolute difference of distances to two foci equals a constant). The minimum value of |z - 1| is achieved when z lies on this hyperbola closest to the point 1.
<p><strong>Step 1:</strong> Interpret the constraint. The equation ||z - 3| - |z + 1|| = 2 represents a hyperbola with foci F₁ = 3 and F₂ = -1. The distance between foci is |3 - (-1)| = 4.</p><p><strong>Step 2:</strong> For a hyperbola, ||z - 3| - |z + 1|| = 2 means 2a = 2, so a = 1. Since c = 2 (half the distance between foci), the vertices lie at z = 1 and z = -3.</p><p><strong>Step 3:</strong> For points on a hyperbola ||z - 3| - |z + 1|| = 2, the point closest to z = 1 is the right vertex itself, which occurs at z = 1.</p><p><strong>Step 4:</strong> Verify: At z = 1: ||1 - 3| - |1 + 1|| = ||−2| - |2|| = |2 - 2| = 0 ✗. Reconsider: the right branch has |z - 3| - |z + 1| = -2 (since z is closer to 3). Points satisfying this with z ≥ 1 form the right branch. The closest point on this branch to 1 is the vertex at z = 1.</p><p><strong>Step 5:</strong> Actually, checking the hyperbola equation more carefully: vertices are at c ± a from origin = 2 ± 1, giving z = 1 and z = 3 on the real axis... Correction: the center is at (3-1)/2 = 1, vertices at 1 ± 1. The right branch vertex at z = 2 gives ||2-3| - |2+1|| = |1 - 3| = 2 ✓.</p><p><strong>Step 6:</strong> The minimum |z - 1| for points on this hyperbola occurs at z = 2 (the closest point on the right branch to 1). Thus |2 - 1| = <strong>1</strong>.</p>
Correct Answer: 1