<p>The equation \(14x^2 - 4xy + 11y^2 - 44x - 58y + 71 = 0\) whose centre is</p>
Step-by-Step Solution
Key Concept: Transform the general second-degree equation into standard form by completing the square after rotating axes to eliminate the xy term, or find the center by treating it as a conic and using the condition that partial derivatives equal zero at the center.
<p><strong>Step 1:</strong> For a general conic equation f(x,y) = 14x² - 4xy + 11y² - 44x - 58y + 71 = 0, the center (h,k) is found using:</p><p>∂f/∂x = 0 and ∂f/∂y = 0</p><p><strong>Step 2:</strong> Calculate partial derivatives:</p><p>∂f/∂x = 28x - 4y - 44 = 0 ... (1)</p><p>∂f/∂y = -4x + 22y - 58 = 0 ... (2)</p><p><strong>Step 3:</strong> From equation (1): 28x - 4y = 44 → 7x - y = 11</p><p>From equation (2): -4x + 22y = 58 → -2x + 11y = 29</p><p><strong>Step 4:</strong> Solve the system:</p><p>From first: y = 7x - 11</p><p>Substitute into second: -2x + 11(7x - 11) = 29</p><p>-2x + 77x - 121 = 29</p><p>75x = 150</p><p>x = 2</p><p><strong>Step 5:</strong> y = 7(2) - 11 = 3</p><p>∴ Center = (2, 3)</p>
Correct Answer: A