<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: Parametrize the constraint x² + y² = 4 using x = 2cosθ, y = 2sinθ, then express x² - xy + y² in terms of θ to find its range analytically.
<p><strong>Step 1:</strong> Since x² + y² = 4, parametrize using x = 2cosθ, y = 2sinθ where θ ∈ [0, 2π).</p><p><strong>Step 2:</strong> Substitute into the expression:<br/>x² - xy + y² = 4cos²θ - (2cosθ)(2sinθ) + 4sin²θ<br/>= 4(cos²θ + sin²θ) - 4cosθsinθ<br/>= 4 - 4cosθsinθ<br/>= 4 - 2sin(2θ)</p><p><strong>Step 3:</strong> Find the range. Since sin(2θ) ∈ [-1, 1]:<br/>• Maximum: when sin(2θ) = -1, we get 4 - 2(-1) = 6<br/>• Minimum: when sin(2θ) = 1, we get 4 - 2(1) = 2</p><p><strong>Step 4:</strong> Therefore, x² - xy + y² ∈ [2, 6].</p><p><strong>Step 5:</strong> The integers in the range [2, 6] are: 2, 3, 4, 5, 6.</p><p>∴ Answer: <strong>5 integers</strong></p>
Correct Answer: 5