<p>The minimum distance of a point on the curve \(y = x^2 - 4\) from the origin is</p>
<p>\(\dfrac{\sqrt{15}}{2}\)</p>
<p>\(\sqrt{\dfrac{19}{2}}\)</p>
<p>\(\sqrt{\dfrac{15}{2}}\)</p>
<p>\(\dfrac{\sqrt{19}}{2}\)</p>
Step-by-Step Solution
Key Concept: The distance from origin to a point (x, x²-4) on the curve is √(x² + (x²-4)²). Minimizing this is equivalent to minimizing the square of distance D² = x² + (x²-4)², which eliminates the square root and simplifies differentiation.
<p><strong>Step 1:</strong> Let P(x, x²-4) be a point on the curve y = x² - 4. The distance from origin is D = √[x² + (x²-4)²]</p><p><strong>Step 2:</strong> To minimize D, minimize D² = x² + (x²-4)² = x² + x⁴ - 8x² + 16 = x⁴ - 7x² + 16</p><p><strong>Step 3:</strong> Differentiate: d(D²)/dx = 4x³ - 14x = 2x(2x² - 7) = 0</p><p>This gives x = 0 or x² = 7/2, so x = 0 or x = ±√(7/2)</p><p><strong>Step 4:</strong> Evaluate D² at critical points:</p><p>• At x = 0: D² = 0 + 16 = 16, so D = 4</p><p>• At x² = 7/2: D² = (7/2)² - 7(7/2) + 16 = 49/4 - 49/2 + 16 = 49/4 - 98/4 + 64/4 = 15/4, so D = √(15/4) = √15/2</p><p><strong>Step 5:</strong> Check second derivative: d²(D²)/dx² = 12x² - 14. At x² = 7/2: d²(D²)/dx² = 12(7/2) - 14 = 42 - 14 = 28 > 0 ✓ (minimum)</p><p>Since √15/2 ≈ 1.94 < 4, the minimum distance is √15/2</p><p>∴ Answer: C</p>
Correct Answer: C