<p>The sum of maximum and minimum modulus of a complex number <i>z</i> satisfying <i>|z - 25i| ≤ 15</i>, where <i>i = √(-1)</i> is <i>S</i>. Then <i>S/10</i> is:</p>
Step-by-Step Solution
Key Concept: The constraint |z - 25i| ≤ 15 represents a closed disk centered at 25i with radius 15. The modulus |z| represents the distance from the origin, so we need to find the maximum and minimum distances from the origin to points in this disk.
<p><strong>Step 1:</strong> Interpret the constraint |z - 25i| ≤ 15</p><p>This inequality represents all complex numbers z lying in or on a closed disk centered at the point 25i with radius r = 15.</p><p><strong>Step 2:</strong> Find the center and radius</p><p>Center: c = 25i (located on the imaginary axis)</p><p>Radius: R = 15</p><p><strong>Step 3:</strong> Determine position of origin relative to disk</p><p>Distance from origin O to center c = |25i - 0| = |25i| = 25</p><p>Since 25 > 15, the origin lies outside the disk.</p><p><strong>Step 4:</strong> Find maximum modulus</p><p>The maximum value of |z| occurs at the point on the disk farthest from the origin, which lies on the line from O through c, on the far side of the disk.</p><p>|z|<sub>max</sub> = distance from O to c + radius = 25 + 15 = 40</p><p><strong>Step 5:</strong> Find minimum modulus</p><p>The minimum value of |z| occurs at the point on the disk closest to the origin, which lies on the line from O through c, on the near side of the disk.</p><p>|z|<sub>min</sub> = distance from O to c - radius = 25 - 15 = 10</p><p><strong>Step 6:</strong> Calculate S and S/10</p><p>S = |z|<sub>max</sub> + |z|<sub>min</sub> = 40 + 10 = 50</p><p>S/10 = 50/10 = 5</p><p><strong>∴ Answer:</strong> 5</p>
Correct Answer: 5