<p>The solutions of equations \(x^2 + y^2 - 4x - 4y = 60\) and \(xy + 2x + 2y = 20\) satisfy the following equation(s).</p>
Step-by-Step Solution
Key Concept: Rewrite both equations by completing the square and factoring to reveal that (x-2)(y-2) and (x+2)(y+2) are related, allowing you to substitute u = x+y and v = xy to solve systematically.
<p><strong>Step 1:</strong> Rewrite the first equation by completing the square:</p><p>x² + y² - 4x - 4y = 60</p><p>(x² - 4x + 4) + (y² - 4y + 4) - 8 = 60</p><p>(x - 2)² + (y - 2)² = 68</p><p><strong>Step 2:</strong> Rewrite the second equation by factoring:</p><p>xy + 2x + 2y = 20</p><p>x(y + 2) + 2(y + 2) = 20 + 4</p><p>(x + 2)(y + 2) = 24</p><p><strong>Step 3:</strong> Let u = x + 2 and v = y + 2. Then uv = 24 and</p><p>(u - 4)² + (v - 4)² = 68</p><p>u² - 8u + 16 + v² - 8v + 16 = 68</p><p>u² + v² - 8(u + v) = 36</p><p><strong>Step 4:</strong> Since uv = 24, we have u² + v² = (u + v)² - 2uv = (u + v)² - 48</p><p>(u + v)² - 48 - 8(u + v) = 36</p><p>(u + v)² - 8(u + v) - 84 = 0</p><p>Let s = u + v: s² - 8s - 84 = 0</p><p>(s - 14)(s + 6) = 0, so s = 14 or s = -6</p><p><strong>Step 5:</strong> This means x + y + 4 = 14 or x + y + 4 = -6</p><p>∴ x + y = 10 or x + y = -10 (Answer: A)</p>
Correct Answer: A