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If the straight line y=x meets y=f(x) at...

If the straight line `y=x` meets `y=f(x)` at P, where `f(x)` is a solution of the differential equation `(dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2))` such that `f(1)=3`, then the value of `f'(x)` at the point P is

A

`(3)/(5)`

B

`(5)/(3)`

C

2

D

1

Text Solution

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The correct Answer is:
To solve the problem, we need to find the value of \( f'(x) \) at the point \( P \) where the line \( y = x \) meets the curve \( y = f(x) \), given that \( f(x) \) is a solution to the differential equation: \[ \frac{dy}{dx} = \frac{x^2 + xy}{x^2 + y^2} \] with the condition \( f(1) = 3 \). ### Step-by-step Solution: 1. **Understand the Intersection Point**: Since the line \( y = x \) meets the curve \( y = f(x) \) at point \( P \), at this point, we have: \[ y = f(x) = x \] Therefore, at point \( P \), \( x = y \). 2. **Substitute into the Differential Equation**: We can substitute \( y = x \) into the differential equation: \[ \frac{dy}{dx} = \frac{x^2 + xy}{x^2 + y^2} \] becomes: \[ f'(x) = \frac{x^2 + x \cdot x}{x^2 + x^2} = \frac{x^2 + x^2}{x^2 + x^2} = \frac{2x^2}{2x^2} = 1 \] 3. **Conclusion**: Therefore, at the point \( P \), we find that: \[ f'(x) = 1 \] ### Final Answer: The value of \( f'(x) \) at the point \( P \) is \( 1 \). ---
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