<p>Given line \(y = x\) and curve \(y^2 = x - 2\). The shortest distance between the line and the curve is:</p>
Step-by-Step Solution
Key Concept: The shortest distance from a point on the curve to the line occurs when the tangent to the curve is parallel to the given line. Use the perpendicular distance formula from a point to a line after finding the critical point.
<p><strong>Step 1:</strong> Curve is y² = x - 2, or x = y² + 2. For a point P(y² + 2, y) on the curve, find when the tangent is parallel to y = x.</p><p><strong>Step 2:</strong> Differentiate: dx/dy = 2y, so dy/dx = 1/(2y). For tangent parallel to y = x, we need dy/dx = 1, giving 1/(2y) = 1, so y = 1/2.</p><p><strong>Step 3:</strong> At y = 1/2, the point on curve is P(1/4 + 2, 1/2) = P(9/4, 1/2).</p><p><strong>Step 4:</strong> Distance from point (9/4, 1/2) to line x - y = 0 is: d = |9/4 - 1/2|/√2 = |9/4 - 2/4|/√2 = (7/4)/√2 = 7/(4√2) = 7√2/8 ≈ 1.2374.</p><p>∴ Answer: <strong>1.2374</strong></p>
Correct Answer: 1.2374