<p>\(x^2 - xy + y^2 - 4x - 4y + 16 = 0\) represents</p>
Step-by-Step Solution
Key Concept: Rearrange the equation as a quadratic in x and analyze its discriminant relative to y. When the discriminant equals zero for all y, the equation represents a single point (degenerate conic).
<p><strong>Step 1:</strong> Treat as quadratic in x: x² - x(y + 4) + (y² - 4y + 16) = 0</p><p><strong>Step 2:</strong> Calculate discriminant: Δ = (y + 4)² - 4(y² - 4y + 16) = y² + 8y + 16 - 4y² + 16y - 64 = -3y² + 24y - 48 = -3(y² - 8y + 16) = -3(y - 4)²</p><p><strong>Step 3:</strong> For real solutions, Δ ≥ 0, but -3(y - 4)² ≤ 0 for all y. Equality holds only when y = 4.</p><p><strong>Step 4:</strong> When y = 4: x² - 4x - 4(4) + 16 = 0 → x² - 4x = 0 → x(x - 4) = 0. Wait, recalculate: substitute y = 4 gives x² - 4x + (16 - 16 + 16) = 0 → x² - 4x + 16 = 0. Actually, from discriminant = 0 when y = 4: x = (4 + 4)/2 = 4</p><p><strong>Step 5:</strong> Verify: (4)² - 4(4) + 16 - 4(4) - 4(4) + 16 = 16 - 16 + 16 - 16 - 16 + 16 = 0 ✓</p><p>∴ The equation represents a <strong>single point (4, 4)</strong> or a <strong>degenerate conic</strong></p>
Correct Answer: A