If ` x : y = 7 : 3, ` then the value of ` (xy+y^(2))/(x^(2) - y^(2)) ` is:

Options

A. ` (3)/(4)`
B. ` (4)/(3)`
C. ` (3)/(7)`
D. ` (7)/(3)`

Answer

To solve the problem where \( x : y = 7 : 3 \) and we need to find the value of \( \frac{xy + y^2}{x^2 - y^2} \), we can follow these steps:

### Step 1: Express \( x \) and \( y \) in terms of a variable
Since \( x : y = 7 : 3 \), we can express \( x \) and \( y \) as:
\[ x = 7k \quad \text{and} \quad y = 3k \]
where \( k \) is a common multiplier.

### Step 2: Substitute \( x \) and \( y \) into the expression
Now, substitute \( x \) and \( y \) into the expression \( \frac{xy + y^2}{x^2 - y^2} \):
\[ xy = (7k)(3k) = 21k^2 \]
\[ y^2 = (3k)^2 = 9k^2 \]
\[ x^2 = (7k)^2 = 49k^2 \]
\[ y^2 = 9k^2 \]

### Step 3: Calculate the numerator and denominator
Now, calculate the numerator:
\[ xy + y^2 = 21k^2 + 9k^2 = 30k^2 \]
And the denominator:
\[ x^2 - y^2 = 49k^2 - 9k^2 = 40k^2 \]

### Step 4: Form the complete expression
Now, we can form the complete expression:
\[ \frac{xy + y^2}{x^2 - y^2} = \frac{30k^2}{40k^2} \]

### Step 5: Simplify the expression
The \( k^2 \) terms cancel out:
\[ \frac{30}{40} = \frac{3}{4} \]

### Final Answer
Thus, the value of \( \frac{xy + y^2}{x^2 - y^2} \) is:
\[ \frac{3}{4} \]