<p>Area bounded by the parabola \((y-2)^2 = x - 1\), the tangent to it at the point P (2, 3) and the x-axis is equal to</p>
Step-by-Step Solution
Key Concept: Identify that the area is bounded by three curves: the parabola, its tangent line at P(2,3), and the x-axis. You must find where the tangent intersects the x-axis and parabola, then set up the integral with appropriate limits.
<p><strong>Step 1:</strong> Find the tangent line at P(2, 3). For parabola (y-2)² = x-1, differentiate implicitly: 2(y-2)dy/dx = 1, so dy/dx = 1/[2(y-2)]. At P(2,3): dy/dx = 1/[2(3-2)] = 1/2.</p><p><strong>Step 2:</strong> Tangent line equation: y - 3 = (1/2)(x - 2) → y = (1/2)x + 2. Find x-intercept: 0 = (1/2)x + 2 → x = -4.</p><p><strong>Step 3:</strong> Express parabola as x = (y-2)² + 1. The parabola passes through P(2,3) and has vertex at (1,2).</p><p><strong>Step 4:</strong> Set up area integral. From x = -4 to x = 1, the tangent line bounds the region above. From x = 1 to x = 2, the parabola bounds it. Area = ∫₋₄¹ [(1/2)x + 2] dx + ∫₁² [(y-2)² + 1] dy (converting to y-integration from y=2 to y=3).</p><p><strong>Step 5:</strong> Calculate: ∫₋₄¹ [(1/2)x + 2] dx = [(1/4)x² + 2x]₋₄¹ = (1/4 + 2) - (4 - 8) = 9/4 + 4 = 25/4.</p><p><strong>Step 6:</strong> For parabola part: ∫₂³ (y-2)² dy = [(y-2)³/3]₂³ = 1/3.</p><p><strong>Step 7:</strong> Total area = 25/4 + 1/3 = 75/12 + 4/12 = 79/12 (or verify with standard answer format)</p><p>∴ Answer: <strong>79/12 or 6 7/12 square units</strong></p>
Correct Answer: 79