<p>Find the locus of the points representing the complex number \(z\) for which \(|z+5|^2 - |z-5|^2 = 10\).</p>
Step-by-Step Solution
Key Concept: Expand |z+5|² and |z-5|² using (z+a)(z̄+ā) formula, then the imaginary parts cancel leaving only real parts. This immediately gives a linear equation in Re(z).
<p><strong>Step 1:</strong> Let z = x + iy where x, y ∈ ℝ.</p><p><strong>Step 2:</strong> Calculate |z+5|² = |(x+5) + iy|² = (x+5)² + y²</p><p><strong>Step 3:</strong> Calculate |z-5|² = |(x-5) + iy|² = (x-5)² + y²</p><p><strong>Step 4:</strong> Expand the difference:<br/>|z+5|² - |z-5|² = [(x+5)² + y²] - [(x-5)² + y²]<br/>= (x+5)² - (x-5)²<br/>= (x² + 10x + 25) - (x² - 10x + 25)<br/>= 20x</p><p><strong>Step 5:</strong> Set equal to 10:<br/>20x = 10<br/>x = 1/2</p><p><strong>Step 6:</strong> Since y can be any real number, the locus is the vertical line Re(z) = 1/2 or x = 1/2 in the complex plane.</p><p><strong>∴ Answer:</strong> x = 1/2 (a vertical line parallel to the imaginary axis)
Correct Answer: x = 1/2