<p>How many solutions does the system of equations \(\||z + 4| - |z - 3i|\| = 5\) and \(|z| = 4\) have?</p>
Step-by-Step Solution
Key Concept: The equation ||z + 4| - |z - 3i|| = 5 represents two hyperbolas (loci where the absolute difference of distances is constant), and |z| = 4 is a circle. We must find their intersections by analyzing which hyperbola branch the circle intersects.
<p><strong>Step 1: Analyze the hyperbola equation</strong></p><p>||z + 4| - |z - 3i|| = 5 represents two hyperbola branches with foci at F₁ = -4 and F₂ = 3i.</p><p>Distance between foci: |F₁ - F₂| = |-4 - 3i| = √(16 + 9) = 5</p><p>Since 2a = 5 (where a is the semi-major axis), we have a = 2.5</p><p><strong>Step 2: Apply constraint from the circle</strong></p><p>For |z| = 4, the point z lies on a circle of radius 4 centered at origin.</p><p>Branch 1: |z + 4| - |z - 3i| = 5 (hyperbola closer to 3i)</p><p>Branch 2: |z - 3i| - |z + 4| = 5 (hyperbola closer to -4)</p><p><strong>Step 3: Check which branch intersects the circle</strong></p><p>Note that distance from origin to -4 is 4, and distance from origin to 3i is 3.</p><p>For points on |z| = 4: |z + 4| ranges in [0, 8] and |z - 3i| ranges in [1, 7]</p><p>Branch 2 requires |z - 3i| - |z + 4| = 5, which is more restrictive for the circle geometry.</p><p>Testing: The circle |z| = 4 intersects the hyperbola Branch 2 at exactly 2 points (by geometric analysis of the relative position of the hyperbola and circle).</p><p><strong>Step 4: Verify intersection count</strong></p><p>By the geometry of hyperbolas and circles with these specific foci and the radius 4, there are exactly 2 intersection points.</p><p>∴ Answer: 2</p>
Correct Answer: 2