<p>Let a curve \(y = f(x)\) pass through (1, 1). A tangent at point \(P(x_1, y_1)\) on the curve meets the x-axis at point \(A\) and y-axis at point \(B\). If point \(P\) divides \(AB\) in the ratio \(1:3\) and \(y(1)=1\), then \(y\left(\frac{1}{2}\right)\) equals:</p>
Step-by-Step Solution
Key Concept: Use the section formula with the given ratio 1:3 to relate the coordinates of A, B, and P(x₁, y₁). This creates two equations involving the slope dy/dx, which forms a separable differential equation.
<p><strong>Step 1:</strong> Identify coordinates. Let A = (a, 0) on x-axis and B = (0, b) on y-axis. Point P(x₁, y₁) divides AB in ratio 1:3.</p><p><strong>Step 2:</strong> Using section formula for P dividing AB in 1:3: <br>x₁ = (1·0 + 3·a)/(1+3) = 3a/4, so <strong>a = 4x₁/3</strong><br>y₁ = (1·b + 3·0)/(1+3) = b/4, so <strong>b = 4y₁</strong></p><p><strong>Step 3:</strong> The tangent line passes through P(x₁, y₁), A(4x₁/3, 0), and B(0, 4y₁). Slope from P to A:<br>dy/dx = (y₁ - 0)/(x₁ - 4x₁/3) = y₁/(-x₁/3) = <strong>-3y₁/x₁</strong></p><p><strong>Step 4:</strong> Form the differential equation:<br>dy/dx = -3y/x<br>Separating variables: dy/y = -3dx/x<br>Integrating: ln|y| = -3ln|x| + C<br>Therefore: <strong>y = k/x³</strong></p><p><strong>Step 5:</strong> Apply initial condition y(1) = 1:<br>1 = k/1³, so k = 1<br>Thus: <strong>y = 1/x³</strong></p><p><strong>Step 6:</strong> Find y(1/2):<br>y(1/2) = 1/(1/2)³ = 1/(1/8) = <strong>8</strong></p><p>∴ Answer: C</p>
Correct Answer: C