Question:
The equation of the curve passing through $(2, \frac{9}{2})$ and having the slope $(1 - \frac{1}{x^2})$ at $(x, y)$ is
The equation of the curve passing through $(2, \frac{9}{2})$ and having the slope $(1 - \frac{1}{x^2})$ at $(x, y)$ is
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Slope of the curve is $\frac{dy}{dx}$. Integrate to find the function.
Updated On: Apr 26, 2026
- $xy = x^2 + 2x + 1$
- $xy = x^2 + x + 2$
- $xy = x^2 + x + 5$
- $xy = x^2 + 2x + 5$
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The Correct Option is B
Solution and Explanation
Step 1: Differential Equation
$\frac{dy}{dx} = 1 - \frac{1}{x^2}$.
Step 2: Integration
$y = \int (1 - x^{-2}) dx = x + \frac{1}{x} + C$.
Step 3: Solve for C
Passes through $(2, 4.5) \implies 4.5 = 2 + 0.5 + C \implies C = 2$.
Equation: $y = x + \frac{1}{x} + 2 \implies xy = x^2 + 2x + 1$.
*Recalculating:* $4.5 = 2.5 + C \to C=2$. $y = x + 1/x + 2 \to xy = x^2 + 2x + 1$.
Final Answer: (A)
$\frac{dy}{dx} = 1 - \frac{1}{x^2}$.
Step 2: Integration
$y = \int (1 - x^{-2}) dx = x + \frac{1}{x} + C$.
Step 3: Solve for C
Passes through $(2, 4.5) \implies 4.5 = 2 + 0.5 + C \implies C = 2$.
Equation: $y = x + \frac{1}{x} + 2 \implies xy = x^2 + 2x + 1$.
*Recalculating:* $4.5 = 2.5 + C \to C=2$. $y = x + 1/x + 2 \to xy = x^2 + 2x + 1$.
Final Answer: (A)
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