<p>We have <br/> \( S = \dfrac{\alpha + i}{\alpha - i} \) <br/> If \( S = x + iy \), find \( x^2 + y^2 \).</p>
Step-by-Step Solution
Key Concept: When a complex number is expressed as x + iy, the modulus squared |S|² = x² + y² can be computed directly from the original form using |numerator|²/|denominator|², avoiding algebraic expansion entirely.
<p><strong>Step 1:</strong> Recognize that if S = x + iy, then x² + y² = |S|².</p><p><strong>Step 2:</strong> Use the property that for any complex numbers z and w: |z/w|² = |z|²/|w|².</p><p><strong>Step 3:</strong> Calculate |α + i|² = α² + 1 and |α - i|² = α² + 1.</p><p><strong>Step 4:</strong> Therefore, |S|² = (α² + 1)/(α² + 1) = 1.</p><p><strong>Step 5:</strong> Thus x² + y² = 1.</p><p>∴ Answer: 1</p>
Correct Answer: 1