If x,y are real numbers such that `3^((x)/(y)+1) -3^((x)/(y)-1) = 24`, then the value of `(x+y)//(x-y)` is:-

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( 3^{\frac{x}{y}+1} - 3^{\frac{x}{y}-1} = 24 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3^{\frac{x}{y}+1} - 3^{\frac{x}{y}-1} = 24 \]
This can be rewritten using the properties of exponents:
\[ 3^{\frac{x}{y}} \cdot 3^1 - 3^{\frac{x}{y}} \cdot 3^{-1} = 24 \]
This simplifies to:
\[ 3 \cdot 3^{\frac{x}{y}} - \frac{3^{\frac{x}{y}}}{3} = 24 \]

### Step 2: Factor out \( 3^{\frac{x}{y}} \)
Let \( t = 3^{\frac{x}{y}} \). Then we can rewrite the equation as:
\[ 3t - \frac{t}{3} = 24 \]
To eliminate the fraction, multiply the entire equation by 3:
\[ 9t - t = 72 \]
This simplifies to:
\[ 8t = 72 \]

### Step 3: Solve for \( t \)
Now, divide both sides by 8:
\[ t = \frac{72}{8} = 9 \]

### Step 4: Relate \( t \) back to \( x \) and \( y \)
Since \( t = 3^{\frac{x}{y}} \), we have:
\[ 3^{\frac{x}{y}} = 9 \]
Recognizing that \( 9 = 3^2 \), we can equate the exponents:
\[ \frac{x}{y} = 2 \]

### Step 5: Find \( \frac{x+y}{x-y} \)
From \( \frac{x}{y} = 2 \), we can express \( x \) in terms of \( y \):
\[ x = 2y \]
Now we can find \( x+y \) and \( x-y \):
\[ x+y = 2y + y = 3y \]
\[ x-y = 2y - y = y \]
Now, we can find the ratio:
\[ \frac{x+y}{x-y} = \frac{3y}{y} = 3 \]

### Final Answer
Thus, the value of \( \frac{x+y}{x-y} \) is:
\[ \boxed{3} \]