<p>If the tangent to the curve, \(y = x^3 + ax - b\) at the point \((1, -5)\) is perpendicular to the line, \(-x + y + 4 = 0\), then which one of the following points lie on the curve?</p>
Step-by-Step Solution
Key Concept: Use two conditions simultaneously: (1) the point (1, -5) lies on the curve to find a relationship between a and b, and (2) the tangent at this point is perpendicular to the given line to find the slope constraint, which yields another equation.
<p><strong>Step 1: Find the slope of the given line.</strong></p><p>The line -x + y + 4 = 0 can be rewritten as y = x - 4, so its slope is m₁ = 1.</p><p><strong>Step 2: Find the slope of the tangent to the curve.</strong></p><p>For y = x³ + ax - b, we have dy/dx = 3x² + a.</p><p>At x = 1: dy/dx|ₓ₌₁ = 3(1)² + a = 3 + a</p><p><strong>Step 3: Apply the perpendicularity condition.</strong></p><p>If the tangent is perpendicular to the given line: m_tangent × m_line = -1</p><p>(3 + a)(1) = -1</p><p>Therefore: 3 + a = -1, so <strong>a = -4</strong></p><p><strong>Step 4: Use the point condition to find b.</strong></p><p>The curve passes through (1, -5):</p><p>-5 = (1)³ + a(1) - b</p><p>-5 = 1 + (-4) - b</p><p>-5 = -3 - b</p><p>Therefore: <strong>b = 2</strong></p><p><strong>Step 5: Write the curve equation and verify.</strong></p><p>The curve is: y = x³ - 4x - 2</p><p>Check at (1, -5): y = 1 - 4 - 2 = -5 ✓</p><p>Check tangent slope: dy/dx|ₓ₌₁ = 3 - 4 = -1, and (-1)(1) = -1 ✓</p><p>∴ Answer: <strong>D</strong> (The specific point from options that satisfies y = x³ - 4x - 2)</p>
Correct Answer: D