<p>If <math>|z| = 2</math>, the points representing the complex numbers <math>-1 + 5z</math> will lie on</p>
Step-by-Step Solution
Key Concept: If |z| = 2, then z lies on a circle of radius 2 centered at origin. The expression w = -1 + 5z represents a linear transformation (scaling by 5 and translation by -1), which transforms the circle |z| = 2 into another circle. We must find the locus of w by determining its modulus relationship.
<p><strong>Step 1:</strong> Given that |z| = 2, we know z lies on a circle of radius 2 centered at the origin.</p><p><strong>Step 2:</strong> Let w = -1 + 5z be the complex number we're analyzing. We can rearrange this as: w + 1 = 5z, or equivalently, z = (w + 1)/5</p><p><strong>Step 3:</strong> Since |z| = 2, we substitute: |(w + 1)/5| = 2</p><p><strong>Step 4:</strong> Simplifying using properties of modulus: |w + 1|/|5| = 2, which gives |w + 1|/5 = 2</p><p><strong>Step 5:</strong> Therefore: |w + 1| = 10</p><p><strong>Step 6:</strong> This is the equation of a circle with center at w = -1 (in the complex plane) and radius 10.</p><p><strong>Step 7:</strong> In terms of the original variable, the locus of points representing w = -1 + 5z is a circle centered at (-1, 0) with radius 10.</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A