If the differenential equation `(dx)/(3y+f)+(dy)/(px+g)=0` represents a family of circle, then p=

Options

A. g
B. f
C. 4
D. 3

Explanation

To solve the differential equation

\[ \frac{dx}{3y + f} + \frac{dy}{px + g} = 0 \]

and determine the value of \( p \) such that it represents a family of circles, we can follow these steps:

### Step 1: Rewrite the Differential Equation
We start by rewriting the equation in a more manageable form:

\[ \frac{dx}{3y + f} + \frac{dy}{px + g} = 0 \]

This can be rearranged to:

\[ (px + g)dx + (3y + f)dy = 0 \]

### Step 2: Integrate Both Sides
Next, we will integrate both sides. We can express the left-hand side as:

\[ (px + g)dx + (3y + f)dy = 0 \]

Integrating gives us:

\[ \int (px + g)dx + \int (3y + f)dy = C \]

Calculating the integrals, we have:

\[ \frac{p}{2}x^2 + gx + \frac{3}{2}y^2 + fy = C \]

### Step 3: Identify the Family of Circles
For the equation to represent a family of circles, the coefficients of \( x^2 \) and \( y^2 \) must be equal. Thus, we set:

\[ \frac{p}{2} = \frac{3}{2} \]

### Step 4: Solve for \( p \)
From the equation above, we can solve for \( p \):

\[ p = 3 \]

### Conclusion
Thus, the value of \( p \) is:

\[ \boxed{3} \]