<p>If \(z_1\) lies on \(|z-3| + |z+3| = 8\) such that \(\arg z_1 = \pi/6\), then \(37|z_1|^2 =\) ___.</p>
Step-by-Step Solution
Key Concept: The locus |z-3| + |z+3| = 8 is an ellipse with foci at (±3,0) and semi-major axis a=4. For z₁ with arg(z₁)=π/6, express z₁ in polar form and use the ellipse equation to find |z₁|.
<p><strong>Step 1:</strong> Recognize that |z-3| + |z+3| = 8 is an ellipse with foci F₁=(-3,0) and F₂=(3,0). Sum of distances = 8, so 2a = 8, thus a = 4. Distance between foci = 6, so 2c = 6, thus c = 3.</p><p><strong>Step 2:</strong> Find b: b² = a² - c² = 16 - 9 = 7, so b = √7.</p><p><strong>Step 3:</strong> Since arg(z₁) = π/6, write z₁ = r(cos π/6 + i sin π/6) = r(√3/2 + i/2) where r = |z₁|.</p><p><strong>Step 4:</strong> For a point on an ellipse in parametric form with arg θ, use the focal radius formula. Alternatively, convert z₁ = x + iy where x = r√3/2 and y = r/2, and substitute into the ellipse equation:</p><p><strong>Step 5:</strong> The ellipse equation is x²/16 + y²/7 = 1. Substitute: (r√3/2)²/16 + (r/2)²/7 = 1</p><p>(3r²/4)/16 + (r²/4)/7 = 1</p><p>3r²/64 + r²/28 = 1</p><p><strong>Step 6:</strong> Find common denominator (LCD = 448): (3r² × 7)/(448) + (r² × 16)/(448) = 1</p><p>(21r² + 16r²)/448 = 1</p><p>37r²/448 = 1</p><p>37|z₁|² = 448</p><p>∴ Answer: <strong>448</strong></p>
Correct Answer: 448