If `(x+ y): (x- y) = 3 : 2`, the ratio `(x^2 + y^2) : (x^2 - y^2)` is equal to

Options

A. `5:12`
B. `13:12`
C. `12:13`
D. `24:5`

Answer

To solve the problem, we need to find the ratio \((x^2 + y^2) : (x^2 - y^2)\) given that \((x + y) : (x - y) = 3 : 2\).

### Step-by-step Solution:

1. **Set up the initial ratio**:
\[ \frac{x + y}{x - y} = \frac{3}{2} \]

2. **Cross-multiply to eliminate the fraction**:
\[ 2(x + y) = 3(x - y) \]

3. **Expand both sides**:
\[ 2x + 2y = 3x - 3y \]

4. **Rearrange the equation to isolate terms involving \(x\) and \(y\)**:
\[ 2y + 3y = 3x - 2x \]
\[ 5y = x \]

5. **Express \(x\) in terms of \(y\)**:
\[ x = 5y \]

6. **Substitute \(x\) into the expressions for \(x^2 + y^2\) and \(x^2 - y^2\)**:
\[ x^2 + y^2 = (5y)^2 + y^2 = 25y^2 + y^2 = 26y^2 \]
\[ x^2 - y^2 = (5y)^2 - y^2 = 25y^2 - y^2 = 24y^2 \]

7. **Set up the ratio of \(x^2 + y^2\) to \(x^2 - y^2\)**:
\[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{26y^2}{24y^2} \]

8. **Simplify the ratio**:
\[ \frac{26y^2}{24y^2} = \frac{26}{24} = \frac{13}{12} \]

9. **Final result**:
The ratio \((x^2 + y^2) : (x^2 - y^2)\) is:
\[ 13 : 12 \]