<p>We have \(S = \dfrac{\alpha + i}{\alpha - i}\) where \(\alpha\) is a real number. Then \(x^2 + y^2\) (where \(S = x + iy\)) equals:</p>
Step-by-Step Solution
Key Concept: Multiply numerator and denominator by the conjugate of the denominator to convert S into the form x + iy, then use |S|² = x² + y² = |numerator|²/|denominator|².
<p><strong>Step 1:</strong> Recognize that for S = (α + i)/(α - i), we need x² + y² where S = x + iy.</p><p><strong>Step 2:</strong> Note that x² + y² = |S|², the squared modulus of S.</p><p><strong>Step 3:</strong> Calculate |S|² = |α + i|²/|α - i|² = (α² + 1)/(α² + 1) = 1.</p><p><strong>Verification:</strong> Expanding S = (α + i)(α + i)/[(α - i)(α + i)] = (α² - 1 + 2αi)/(α² + 1), so x = (α² - 1)/(α² + 1) and y = 2α/(α² + 1). Then x² + y² = [(α² - 1)² + 4α²]/(α² + 1)² = (α⁴ + 2α² + 1)/(α² + 1)² = (α² + 1)²/(α² + 1)² = 1.</p><p>∴ Answer: <strong>x² + y² = 1</strong></p>
Correct Answer: B