If `(x^(2)+y^(2))/(x^(2)-y^(2))=17/(8)`, using the properties of proportion find the value of :
(i) x : y

Answer

To solve the equation \(\frac{x^2 + y^2}{x^2 - y^2} = \frac{17}{8}\) and find the value of \(x : y\), we can follow these steps:

### Step-by-Step Solution:

1. **Cross-Multiply the Proportion**:
\[ 8(x^2 + y^2) = 17(x^2 - y^2) \]

2. **Expand Both Sides**:
\[ 8x^2 + 8y^2 = 17x^2 - 17y^2 \]

3. **Rearrange the Equation**:
Move all terms involving \(x^2\) to one side and all terms involving \(y^2\) to the other side:
\[ 8y^2 + 17y^2 = 17x^2 - 8x^2 \]
This simplifies to:
\[ 25y^2 = 9x^2 \]

4. **Divide Both Sides by \(y^2\)**:
\[ \frac{x^2}{y^2} = \frac{25}{9} \]

5. **Take the Square Root of Both Sides**:
\[ \frac{x}{y} = \frac{5}{3} \]

6. **Express the Ratio**:
Therefore, the ratio \(x : y\) is:
\[ x : y = 5 : 3 \]

### Final Answer:
Thus, the value of \(x : y\) is \(5 : 3\).

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