<p>A point on the parabola \(y^2 = 18x\) at which the ordinate increases at twice the rate of the abscissa is</p>
<p>\((2, 4)\)</p>
<p>\((2, -4)\)</p>
<p>\(\left(\dfrac{-9}{8}, \dfrac{9}{2}\right)\)</p>
<p>\(\left(\dfrac{9}{8}, \dfrac{9}{2}\right)\)</p>
Step-by-Step Solution
Key Concept: Use parametric differentiation: if a point moves on the parabola such that dy/dt = 2(dx/dt), we need to find where dy/dx = 2. Since y² = 18x, differentiating implicitly gives 2y(dy/dx) = 18, so dy/dx = 9/y. Setting this equal to 2 yields y = 9/2.
<p><strong>Step 1:</strong> Interpret the condition. If ordinate increases at twice the rate of abscissa, then dy/dt = 2(dx/dt), which gives dy/dx = 2.</p><p><strong>Step 2:</strong> Differentiate y² = 18x implicitly with respect to x: 2y(dy/dx) = 18, so dy/dx = 9/y.</p><p><strong>Step 3:</strong> Set dy/dx = 2: 9/y = 2, which gives y = 9/2.</p><p><strong>Step 4:</strong> Substitute y = 9/2 into the parabola equation: (9/2)² = 18x, so 81/4 = 18x, giving x = 9/8.</p><p><strong>Step 5:</strong> The point is (9/8, 9/2).</p><p>∴ Answer: D</p>
Correct Answer: D