<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 coordinates, same function value, and same derivative (slope). These three equations uniquely determine the three unknowns a, b, and c.
<p><strong>Step 1: Use the point (1,0) lies on both curves</strong></p><p>For g(x): g(1) = 1 + a + b = 0 → a + b = -1 ... (i)</p><p>For h(x): h(1) = c - 1 = 0 → c = 1 ... (ii)</p><p><strong>Step 2: Use the condition that both have the same tangent at (1,0)</strong></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 identical: g'(1) = h'(1)</p><p>2 + a = c - 2</p><p>2 + a = 1 - 2 (using c = 1 from step 1)</p><p>a = -3 ... (iii)</p><p><strong>Step 3: Find b from equation (i)</strong></p><p>a + b = -1</p><p>-3 + b = -1</p><p>b = 2</p><p><strong>Step 4: Calculate (b + c - a)</strong></p><p>b + c - a = 2 + 1 - (-3) = 2 + 1 + 3 = 6</p><p>∴ Answer: <strong>6</strong></p>
Correct Answer: 6