If (x+y): (x-y)= 4:1, then find the ratio `(x^(2)+y^(2)):(x^(2)-y^(2))`

Answer

To solve the problem, we need to find the ratio \((x^2 + y^2) : (x^2 - y^2)\) given that \((x + y) : (x - y) = 4 : 1\).

### Step-by-Step Solution:

1. **Set up the ratio**:
\[ \frac{x + y}{x - y} = \frac{4}{1} \]

2. **Cross-multiply to eliminate the fraction**:
\[ x + y = 4(x - y) \]

3. **Expand the right side**:
\[ x + y = 4x - 4y \]

4. **Rearrange the equation**:
\[ y + 4y = 4x - x \]
\[ 5y = 3x \]

5. **Express \(x\) in terms of \(y\)**:
\[ \frac{x}{y} = \frac{5}{3} \quad \Rightarrow \quad x = \frac{5}{3}y \]

6. **Substitute \(x\) in the expression \(x^2 + y^2\)**:
\[ x^2 + y^2 = \left(\frac{5}{3}y\right)^2 + y^2 = \frac{25}{9}y^2 + y^2 \]
\[ = \frac{25}{9}y^2 + \frac{9}{9}y^2 = \frac{34}{9}y^2 \]

7. **Substitute \(x\) in the expression \(x^2 - y^2\)**:
\[ x^2 - y^2 = \left(\frac{5}{3}y\right)^2 - y^2 = \frac{25}{9}y^2 - y^2 \]
\[ = \frac{25}{9}y^2 - \frac{9}{9}y^2 = \frac{16}{9}y^2 \]

8. **Now, find the ratio \((x^2 + y^2) : (x^2 - y^2)\)**:
\[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{\frac{34}{9}y^2}{\frac{16}{9}y^2} \]

9. **Cancel \(y^2\) and simplify**:
\[ = \frac{34}{16} = \frac{17}{8} \]

10. **Final answer**:
The ratio \((x^2 + y^2) : (x^2 - y^2) = 17 : 8\).