<p>Let \(x\) and \(y\) are real numbers satisfying \(x^2 + y^2 = 4\), then find the number of integers in the range of \((x^2 - xy + y^2)\).</p>
Step-by-Step Solution
Key Concept: Express x² - xy + y² in terms of the constraint x² + y² = 4 by substituting and using parametric form or algebraic manipulation to find its range.
<p><strong>Step 1:</strong> Use the constraint x² + y² = 4. We can write x² - xy + y² = (x² + y²) - xy = 4 - xy.</p><p><strong>Step 2:</strong> Find the range of xy. Parametrize using x = 2cosθ, y = 2sinθ, so xy = 4cosθsinθ = 2sin(2θ).</p><p><strong>Step 3:</strong> The range of 2sin(2θ) is [-2, 2], so xy ∈ [-2, 2].</p><p><strong>Step 4:</strong> Therefore, x² - xy + y² = 4 - xy ranges from 4 - 2 = 2 to 4 - (-2) = 6, giving the range [2, 6].</p><p><strong>Step 5:</strong> Verify attainability: At xy = 2 (when θ = π/4), we get value 2. At xy = -2 (when θ = 3π/4), we get value 6. All intermediate values are attained by continuity.</p><p><strong>Step 6:</strong> The integers in [2, 6] are: 2, 3, 4, 5, 6.</p><p>∴ Answer: <strong>5 integers</strong></p>
Correct Answer: 5