If the solution of the differential equation `(dy)/(dx ) =( ax+ 4) /( 2y + f)`represents a circle, then the value of a is:

Options

A. 2
B. `-2`
C. `3`
D. `-3`

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{ax + 4}{2y + f}\) and find the value of \(a\) such that the solution represents a circle, follow these steps:

### Step 1: Cross-Multiply
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{ax + 4}{2y + f} \]
Cross-multiplying gives us:
\[ (2y + f) \, dy = (ax + 4) \, dx \]

### Step 2: Integrate Both Sides
Now we integrate both sides:
\[ \int (2y + f) \, dy = \int (ax + 4) \, dx \]
The left side integrates to:
\[ y^2 + fy \]
And the right side integrates to:
\[ \frac{a}{2}x^2 + 4x + C \]
Thus, we have:
\[ y^2 + fy = \frac{a}{2}x^2 + 4x + C \]

### Step 3: Rearranging the Equation
Rearranging gives us:
\[ y^2 + fy - \frac{a}{2}x^2 - 4x - C = 0 \]

### Step 4: Identify the Circle Equation
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
To match this form, we need to rearrange our equation:
\[ -\frac{a}{2}x^2 + y^2 + fy - 4x - C = 0 \]
This can be rewritten as:
\[ \frac{-a}{2}x^2 + y^2 - 4x + fy - C = 0 \]

### Step 5: Compare Coefficients
For the equation to represent a circle, the coefficients of \(x^2\) and \(y^2\) must satisfy certain conditions. Specifically, the coefficient of \(x^2\) must be negative and equal to the coefficient of \(y^2\):
\[ -\frac{a}{2} = 1 \quad \text{(since the coefficient of } y^2 \text{ is } 1\text{)} \]
From this, we can solve for \(a\):
\[ -\frac{a}{2} = 1 \implies a = -2 \]

### Conclusion
Thus, the value of \(a\) is:
\[ \boxed{-2} \]