<p>How many solutions does the system of equations \(\arg(z + 2 - 3i) = -\pi/4\) and \(|z + 4| - |z - 3i| = 5\) have?</p>
Step-by-Step Solution
Key Concept: The first equation represents a ray from point (-2, 3) at angle -π/4, while the second represents a hyperbola branch defined by the difference of distances. These two loci must intersect for solutions to exist.
<p><strong>Step 1:</strong> Interpret arg(z + 2 - 3i) = -π/4. This represents a ray starting from point (-2, 3) in the direction making angle -π/4 with the positive real axis. The ray equation is: y - 3 = -1(x + 2), or <strong>y = -x + 1</strong> (for x ≥ -2).</p><p><strong>Step 2:</strong> Interpret |z + 4| - |z - 3i| = 5. Let z = x + iy. The foci are F₁ = (-4, 0) and F₂ = (0, 3). The distance between foci is √(16 + 9) = 5. Since the constant difference equals the distance between foci, this is a degenerate hyperbola—specifically, the ray from F₂ away from F₁.</p><p><strong>Step 3:</strong> For the hyperbola |z + 4| - |z - 3i| = 5, points satisfy this when they lie on the ray from (0, 3) extending in the direction away from (-4, 0). This ray has the equation: y = 3 + (3/4)x for x ≥ 0.</p><p><strong>Step 4:</strong> Find intersection of y = -x + 1 (x ≥ -2) and y = 3 + (3/4)x (x ≥ 0). Setting equal: -x + 1 = 3 + (3/4)x → -(7/4)x = 2 → x = -8/7. Since x = -8/7 < 0, this violates the domain constraint x ≥ 0 for the second ray.</p><p><strong>Step 5:</strong> The two loci do not intersect in valid regions.</p><p>∴ Answer: <strong>No solution</strong></p>
Correct Answer: No solution