<p>If complex number \(z\) lies on the curve \(|z + 1 - i| = 1\), then find the locus of the complex number \(w = \dfrac{z + i}{1 - i}\), where \(i = \sqrt{-1}\).</p>
Step-by-Step Solution
Key Concept: Transform the locus of z into the locus of w by solving z = w(1-i) - i and substituting into the original constraint |z + 1 - i| = 1. The linear transformation preserves circular shapes but scales and rotates them according to the multiplier.
<p><strong>Step 1:</strong> Express z in terms of w. From w = (z+i)/(1-i), we get:</p><p>z = w(1-i) - i</p><p><strong>Step 2:</strong> Substitute into |z + 1 - i| = 1:</p><p>|w(1-i) - i + 1 - i| = 1</p><p>|w(1-i) + 1 - 2i| = 1</p><p><strong>Step 3:</strong> Simplify by factoring. Rewrite as:</p><p>|(1-i)(w + (1-2i)/(1-i))| = 1</p><p><strong>Step 4:</strong> Rationalize (1-2i)/(1-i):</p><p>(1-2i)/(1-i) · (1+i)/(1+i) = (1-2i+i+2)/(2) = (3-i)/2</p><p><strong>Step 5:</strong> Apply |1-i| = √2:</p><p>√2 · |w + (3-i)/2| = 1</p><p>|w + 3/2 - i/2| = 1/√2</p><p><strong>Step 6:</strong> This is a circle with center at w = -3/2 + i/2 and radius 1/√2.</p><p>∴ Answer: A circle with center at (-3/2, 1/2) and radius 1/√2</p>
Correct Answer: A circle with center at (-3/2, 1/2) and radius 1/√2