Complex Numbers
Least modulus
Grade 11
Question:
<p>If \(z\) is a complex number having least modulus and \(z - 2 - 2i| = 1\), then \(z =\)</p>
<p>\((2 - 1/\sqrt{2})(1 - i)\)</p>
<p>\((2 - 1/\sqrt{2})(1 + i)\)</p>
<p>\((2 + 1/\sqrt{2})(1 - i)\)</p>
<p>\((2 + 1/\sqrt{2})(1 + i)\)</p>
Step-by-Step Solution
Key Concept: The complex number with least modulus lying on the circle |z - 2 - 2i| = 1 is the point on this circle closest to the origin. This point lies on the line joining origin to the center (2 + 2i), at distance (radius) from the center toward the origin.
<p><strong>Step 1:</strong> The constraint |z - 2 - 2i| = 1 represents a circle with center C = 2 + 2i and radius r = 1.</p><p><strong>Step 2:</strong> The complex number with least modulus on this circle is the point closest to origin O. This lies on the line from O through C, on the near side of the circle.</p><p><strong>Step 3:</strong> Distance from origin to center: |2 + 2i| = √(4 + 4) = √8 = 2√2</p><p><strong>Step 4:</strong> The minimum modulus point is at distance (2√2 - 1) from origin along the direction of (2 + 2i).</p><p><strong>Step 5:</strong> Direction unit vector: (2 + 2i)/(2√2) = (1 + i)/√2</p><p><strong>Step 6:</strong> Therefore: z = (2√2 - 1) · (1 + i)/√2 = (2√2 - 1)(1 + i)/√2 = √2(2√2 - 1)(1 + i)/2 = (4 - √2)(1 + i)/2 = (2 - √2/2) + (2 - √2/2)i</p><p><strong>Simplified:</strong> z = (2 + 2i) - (1 + i) · (2√2)/(2√2) = (2 + 2i) - (1 + i) = 1 + i (if r=1, scaling gives us the nearest point on circle)</p><p><strong>Correct approach:</strong> z = (2 + 2i) · (2√2 - 1)/(2√2) = (1 + i)(2√2 - 1)/√2 = (1 + i)·√2(√2 - 1/√2) = (1 + i)(2 - √2/√2)</p><p>∴ Answer: <strong>z = (2 - √2)(1 + i)/√2</strong> or equivalently <strong>z = √2(√2 - 1/2)(1 + i)</strong></p>
Correct Answer: A