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 determine the value of \(a\) such that the solution represents a circle, we will follow these steps:

### Step 1: Separate the Variables
We start with the given differential equation:
\[ \frac{dy}{dx} = \frac{ax + 4}{2y + f} \]
We can separate the variables by multiplying both sides by \(2y + f\) and \(dx\):
\[ (2y + f) dy = (ax + 4) dx \]

### Step 2: Integrate Both Sides
Now, we integrate both sides:
\[ \int (2y + f) dy = \int (ax + 4) dx \]
The left side becomes:
\[ \int (2y + f) dy = 2 \cdot \frac{y^2}{2} + fy = y^2 + fy \]
The right side becomes:
\[ \int (ax + 4) dx = \frac{a}{2} x^2 + 4x \]
Thus, we have:
\[ y^2 + fy = \frac{a}{2} x^2 + 4x + C \]
where \(C\) is the constant of integration.

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

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

### Step 5: Compare Coefficients
From our rearranged equation:
- The coefficient of \(x^2\) is \(\frac{a}{2}\)
- The coefficient of \(y^2\) is \(-1\)

Setting these equal gives:
\[ \frac{a}{2} = -1 \]

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

Thus, the value of \(a\) is \(-2\).

### Final Answer
The value of \(a\) is \(-2\).

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