Two numbers x and y `( x gt y)` are such that their sum is equal to three times their difference. Then value of `(3xy)/( 2 (x^(2) - y^(2)))` will be

Options

A. `(2)/(3)`
B. 1
C. `1(1)/(2)`
D. `1(2)/(3)`

Answer

To solve the problem step by step, we start with the given conditions and equations.

### Step 1: Set up the equation based on the problem statement
We know that the sum of the two numbers \( x \) and \( y \) is equal to three times their difference. This can be expressed mathematically as:
\[ x + y = 3(x - y) \]

### Step 2: Expand and simplify the equation
Expanding the right side of the equation gives us:
\[ x + y = 3x - 3y \]
Now, we can rearrange the equation to isolate \( x \) and \( y \):
\[ x + y + 3y = 3x \]
This simplifies to:
\[ x + 4y = 3x \]

### Step 3: Rearranging to find a relationship between \( x \) and \( y \)
Now, we can move \( x \) to the left side:
\[ 4y = 3x - x \]
This simplifies to:
\[ 4y = 2x \]
Dividing both sides by 2 gives:
\[ 2y = x \]
Thus, we can express \( x \) in terms of \( y \):
\[ x = 2y \]

### Step 4: Substitute \( x \) into the expression we need to evaluate
We need to find the value of:
\[ \frac{3xy}{2(x^2 - y^2)} \]
Substituting \( x = 2y \) into this expression:
\[ \frac{3(2y)y}{2((2y)^2 - y^2)} \]

### Step 5: Simplify the expression
Calculating \( (2y)^2 - y^2 \):
\[ (2y)^2 = 4y^2 \quad \text{and} \quad y^2 = y^2 \]
Thus:
\[ (2y)^2 - y^2 = 4y^2 - y^2 = 3y^2 \]
Now substitute this back into the expression:
\[ \frac{3(2y)y}{2(3y^2)} = \frac{6y^2}{6y^2} = 1 \]

### Final Result
The value of \(\frac{3xy}{2(x^2 - y^2)}\) is:
\[ \boxed{1} \]