<p>The minimum distance between the circle \(x^2 + y^2 = 9\) and the curve \(2x^2 + 10y^2 + 6xy = 1\) is:</p>
Step-by-Step Solution
Key Concept: The minimum distance between the circle and curve equals the radius minus the minimum distance from the origin to the curve. We must identify the conic type by diagonalizing the quadratic form to find the closest point on the curve to the origin.
<p><strong>Step 1: Identify the Circle</strong><br/>The circle x² + y² = 9 has center O(0,0) and radius r = 3.</p><p><strong>Step 2: Analyze the Quadratic Form</strong><br/>The curve is 2x² + 10y² + 6xy = 1. Write the matrix form:<br/>A = [2, 3; 3, 10] where Q = x^T A x = 1</p><p><strong>Step 3: Find Eigenvalues</strong><br/>Characteristic equation: det(A - λI) = 0<br/>(2-λ)(10-λ) - 9 = 0<br/>λ² - 12λ + 20 - 9 = 0<br/>λ² - 12λ + 11 = 0<br/>(λ - 1)(λ - 11) = 0<br/>So λ₁ = 1, λ₂ = 11</p><p><strong>Step 4: Transform to Principal Axes</strong><br/>In principal axes, the curve becomes:<br/>u²/1 + v²/(1/11) = 1<br/>This is an ellipse with semi-major axis a = 1 and semi-minor axis b = 1/√11</p><p><strong>Step 5: Find Minimum Distance from Origin to Curve</strong><br/>The closest point on the ellipse to the origin lies along the semi-minor axis (shortest distance):<br/>d_min = b = 1/√11</p><p><strong>Step 6: Calculate Minimum Distance Between Circle and Curve</strong><br/>Minimum distance = (radius of circle) - (minimum distance from O to curve)<br/>= 3 - 1/√11</p><p><strong>∴ Answer:</strong> d</p>
Correct Answer: d