If the solution of the differential equation `(dy)/(dx) = (ax + 3)/(2y + f)` represents a circle , then what is the value of a ?

Options

A. 2
B. 1
C. `-2`
D. `-1`

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{ax + 3}{2y + f}\) and find the value of \(a\) such that the solution represents a circle, we can follow these steps:

### Step 1: Separate Variables
We start by rewriting the differential equation to separate the variables \(y\) and \(x\):
\[ (2y + f) dy = (ax + 3) dx \]

### Step 2: Integrate Both Sides
Next, we integrate both sides:
\[ \int (2y + f) dy = \int (ax + 3) dx \]

The left side integrates to:
\[ y^2 + fy + C_1 \]
And the right side integrates to:
\[ \frac{a}{2}x^2 + 3x + C_2 \]

Thus, we have:
\[ y^2 + fy = \frac{a}{2}x^2 + 3x + C \]
where \(C = C_2 - C_1\).

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

### Step 4: Compare with the General Equation of a Circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
To represent a circle, the coefficients of \(x^2\) and \(y^2\) must be equal and both equal to 1.

From our rearranged equation, we can identify:
- The coefficient of \(y^2\) is \(1\).
- The coefficient of \(x^2\) is \(-\frac{a}{2}\).

### Step 5: Set Coefficients Equal
For the equation to represent a circle, we set:
\[ -\frac{a}{2} = 1 \]

### Step 6: Solve for \(a\)
Multiplying both sides by \(-2\) gives:
\[ a = -2 \]

### Conclusion
Thus, the value of \(a\) such that the solution of the differential equation represents a circle is:
\[ \boxed{-2} \]