<p>Consider the circle \(|z - (4 - 5i)|^2 = 40\) (i.e., centre \((4, -5)\), radius \(= \sqrt{(-4+5i)^2 - (-40)} = \sqrt{4^2 + 5^2 + 40} = 9\)). Which of the following are correct if \(a = \max|z - (-2 + 3i)|\) and \(b = \min|z - (-2 + 3i)|\) for \(z\) on the circle?</p>
Step-by-Step Solution
Key Concept: The maximum and minimum distances from a fixed point to any point on a circle are found by adding/subtracting the radius to the distance between the fixed point and the center. Key: distance from point P(-2, 3i) to center C(4, -5i) determines the baseline.
<p><strong>Step 1: Identify circle parameters</strong></p><p>Circle equation: |z - (4 - 5i)|² = 40</p><p>Centre C = (4, -5i), Radius r = √40 = 2√10</p><p><strong>Step 2: Find distance from P to C</strong></p><p>P = (-2 + 3i), C = (4 - 5i)</p><p>Distance d = |(-2 + 3i) - (4 - 5i)| = |-6 + 8i| = √(36 + 64) = √100 = 10</p><p><strong>Step 3: Calculate maximum distance</strong></p><p>Maximum occurs when z lies on the line from P through C, on the far side:</p><p>a = d + r = 10 + 2√10</p><p><strong>Step 4: Calculate minimum distance</strong></p><p>Minimum occurs when z lies on the line PC, between P and C:</p><p>b = d - r = 10 - 2√10</p><p><strong>Step 5: Verify geometric configuration</strong></p><p>Since d = 10 > r = 2√10 ≈ 6.32, point P lies outside the circle, confirming both max and min formulas apply.</p><p>∴ Answer: Depends on the options provided (typically: a = 10 + 2√10, b = 10 - 2√10, and related relationships)</p>
Correct Answer: a,b,c,d