<p>If the functions \(g(x) = x^2 + ax + b\) and \(h(x) = cx - x^2\) intersect and have the same tangent line at the point \((1, 0)\), then find the value of \((b + c - a)\).</p>
Step-by-Step Solution
Key Concept: At the point of intersection with a common tangent, two curves must satisfy three conditions simultaneously: same point, same function value, and same derivative. Use these three equations to find the unknown coefficients.
<p><strong>Step 1:</strong> Since both curves pass through (1, 0):</p><p>For g(x): g(1) = 1 + a + b = 0 → <strong>a + b = -1</strong> ... (i)</p><p>For h(x): h(1) = c - 1 = 0 → <strong>c = 1</strong> ... (ii)</p><p><strong>Step 2:</strong> Find derivatives and apply equal tangent condition at x = 1:</p><p>g'(x) = 2x + a, so g'(1) = 2 + a</p><p>h'(x) = c - 2x, so h'(1) = c - 2</p><p>Since tangent lines are the same: g'(1) = h'(1)</p><p>2 + a = c - 2 → 2 + a = 1 - 2 → <strong>a = -3</strong> ... (iii)</p><p><strong>Step 3:</strong> From equation (i): -3 + b = -1 → <strong>b = 2</strong></p><p><strong>Step 4:</strong> Calculate b + c - a:</p><p>b + c - a = 2 + 1 - (-3) = 2 + 1 + 3 = <strong>6</strong></p><p>∴ Answer: <strong>6</strong></p>
Correct Answer: 6