Complex Numbers
Modulus and Locus
Grade None
Question:
<p>If <i>z</i> is any complex number satisfying |<i>z</i> - 3 - 2<i>i</i>| ≤ 2, where <i>i</i> = √−1, then the minimum value of |2<i>z</i> - 6 + 5<i>i</i>|, is</p>
Step-by-Step Solution
Key Concept: The minimum distance from a point to a disk equals the distance to the center minus the radius, and we must account for the coefficient 2 in the expression.
<p><strong>Step 1:</strong> The condition |<i>z</i> - 3 - 2<i>i</i>| ≤ 2 represents a closed disk centered at 3 + 2<i>i</i> with radius 2.</p><p><strong>Step 2:</strong> We need to minimize |2<i>z</i> - 6 + 5<i>i</i>| = 2|<i>z</i> - 3 + (5/2)<i>i</i>|.</p><p><strong>Step 3:</strong> The point 3 - (5/2)<i>i</i> lies outside the disk. The minimum distance from this point to the disk is the distance from 3 - (5/2)<i>i</i> to the center 3 + 2<i>i</i> minus the radius 2.</p><p><strong>Step 4:</strong> Distance = |−(5/2)<i>i</i> − 2<i>i</i>| = |(−9/2)<i>i</i>| = 9/2.</p><p><strong>Step 5:</strong> Minimum distance = 9/2 - 2 = 5/2.</p><p>∴ Minimum value of |2<i>z</i> - 6 + 5<i>i</i>| = 2 × (5/2) = <strong>5</strong>.</p>
Correct Answer: 5