A curve `y=f(x)` is passing through (0,0). If slope of the curve at any point (x,y) is equal to (x+xy), then the number of solution of the equation f(x)=1, is :

Options

A. 0
B. 1
C. 2
D. 4

Explanation

To solve the problem, we need to find the number of solutions for the equation \( f(x) = 1 \) given the slope of the curve \( y = f(x) \) at any point \( (x, y) \) is equal to \( x + xy \).

### Step-by-Step Solution:

1. **Identify the slope of the curve**:
We are given that the slope of the curve at any point \( (x, y) \) is \( \frac{dy}{dx} = x + xy \).

2. **Rearranging the equation**:
We can rewrite the equation as:
\[ \frac{dy}{dx} = x(1 + y) \]

3. **Separating variables**:
To solve this differential equation, we separate the variables:
\[ \frac{dy}{1 + y} = x \, dx \]

4. **Integrating both sides**:
Now we integrate both sides:
\[ \int \frac{dy}{1 + y} = \int x \, dx \]
The left side integrates to \( \log(1 + y) \) and the right side integrates to \( \frac{x^2}{2} + C \):
\[ \log(1 + y) = \frac{x^2}{2} + C \]

5. **Using the initial condition**:
We know the curve passes through the point \( (0, 0) \). We can use this to find the constant \( C \):
\[ \log(1 + 0) = \frac{0^2}{2} + C \implies 0 = 0 + C \implies C = 0 \]

6. **Final equation of the curve**:
Substituting \( C \) back into the equation gives us:
\[ \log(1 + y) = \frac{x^2}{2} \]

7. **Finding \( f(x) = 1 \)**:
We need to find the values of \( x \) such that \( f(x) = 1 \). This means:
\[ \log(1 + 1) = \frac{x^2}{2} \]
Simplifying gives:
\[ \log(2) = \frac{x^2}{2} \]
Multiplying both sides by 2:
\[ x^2 = 2 \log(2) \]

8. **Finding the solutions for \( x \)**:
Taking the square root gives:
\[ x = \pm \sqrt{2 \log(2)} \]
This indicates there are two solutions for \( x \).

### Conclusion:
The number of solutions for the equation \( f(x) = 1 \) is **2**.