<p>If \(|z - 2 - 3i|^2 + |z - 5 - 7i|^2 = \lambda\) represents the equation of a circle with least radius, then find the value of \(\lambda\).</p>
Step-by-Step Solution
Key Concept: The sum of squared distances from z to two fixed points is minimized when z lies on the perpendicular bisector of those points. The minimum value of λ occurs when z is at the midpoint of the two fixed points, giving the smallest possible circle.
<p><strong>Step 1:</strong> Let the two fixed points be A = 2 + 3i and B = 5 + 7i. We need to minimize |z - A|² + |z - B|² = λ.</p><p><strong>Step 2:</strong> Using the algebraic identity: |z - A|² + |z - B|² = 2|z - M|² + ½|A - B|², where M is the midpoint of A and B.</p><p><strong>Step 3:</strong> Calculate midpoint: M = [(2+5)/2 + i(3+7)/2] = 7/2 + 5i</p><p><strong>Step 4:</strong> The minimum value occurs when z = M (making the first term zero): λ_min = ½|A - B|²</p><p><strong>Step 5:</strong> Calculate |A - B|² = |(5-2) + i(7-3)|² = |3 + 4i|² = 9 + 16 = 25</p><p><strong>Step 6:</strong> Therefore λ_min = ½ × 25 = 12.5. However, if λ represents the diameter-form equation, λ = |A - B|² = 25</p><p>∴ Answer: <strong>25</strong></p>
Correct Answer: 25