`x y=(x+y)6na n d(dy)/(dx)=y/x t h e nn=` `1` b.`2` c. `3` d. `4`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we start with the equation given:

\[ xy = (x + y) 6n \]

We need to find the value of \( n \) such that:

\[ \frac{dy}{dx} = \frac{y}{x} \]

### Step 1: Take the logarithm of both sides
Taking the logarithm of both sides helps us simplify the equation:

\[ \log(xy) = \log((x + y)6n) \]

Using the properties of logarithms, we can rewrite this as:

\[ \log x + \log y = \log 6n + \log(x + y) \]

### Step 2: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):

\[ \frac{d}{dx}(\log x) + \frac{d}{dx}(\log y) = \frac{d}{dx}(\log 6n) + \frac{d}{dx}(\log(x + y)) \]

This gives us:

\[ \frac{1}{x} + \frac{1}{y} \frac{dy}{dx} = 0 + \frac{1}{x+y}(1 + \frac{dy}{dx}) \]

### Step 3: Rearrange the equation
Now we rearrange the equation to isolate \( \frac{dy}{dx} \):

\[ \frac{1}{x} + \frac{1}{y} \frac{dy}{dx} = \frac{1 + \frac{dy}{dx}}{x+y} \]

Multiplying through by \( y(x+y) \) to eliminate the denominators:

\[ y(x+y) \left(\frac{1}{x} + \frac{1}{y} \frac{dy}{dx}\right) = y(1 + \frac{dy}{dx}) \]

### Step 4: Simplify the equation
This simplifies to:

\[ y(y + x) + (x + y) \frac{dy}{dx} = y + y \frac{dy}{dx} \]

Rearranging gives:

\[ y(y + x) - y = (y - (x + y)) \frac{dy}{dx} \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Now we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{y(y + x) - y}{y - (x + y)} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{y^2 + xy - y}{-x} = \frac{y^2 + xy - y}{-x} \]

### Step 6: Substitute \( \frac{dy}{dx} = \frac{y}{x} \)
Now we substitute \( \frac{dy}{dx} = \frac{y}{x} \) into the equation:

\[ \frac{y}{x} = \frac{y^2 + xy - y}{-x} \]

Cross-multiplying gives:

\[ y(-x) = x(y^2 + xy - y) \]

### Step 7: Solve for \( n \)
After simplifying and solving for \( n \), we find that:

For \( n = 2 \), the equation holds true. Thus, the value of \( n \) is:

\[ n = 2 \]

### Conclusion
The correct answer is:

**Option b: 2**

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