Find the third proportional to `(x)/(y)` + `(y)/(x)` and `sqrt(x^(2) + y^(2))`

Answer

To find the third proportional to \(\frac{x}{y} + \frac{y}{x}\) and \(\sqrt{x^2 + y^2}\), we can follow these steps:

### Step-by-Step Solution

1. **Identify the given terms**:
We have two terms:
- First term: \(A_1 = \frac{x}{y} + \frac{y}{x}\)
- Second term: \(A_2 = \sqrt{x^2 + y^2}\)

2. **Set up the proportion**:
Let the third proportional be \(A\). According to the definition of third proportional, we can set up the following proportion:
\[ \frac{A_1}{A_2} = \frac{A_2}{A} \]
This means:
\[ \frac{\frac{x}{y} + \frac{y}{x}}{\sqrt{x^2 + y^2}} = \frac{\sqrt{x^2 + y^2}}{A} \]

3. **Cross-multiply**:
Cross-multiplying gives us:
\[ A_1 \cdot A = A_2 \cdot A_2 \]
Substituting the values of \(A_1\) and \(A_2\):
\[ \left(\frac{x}{y} + \frac{y}{x}\right) A = \left(\sqrt{x^2 + y^2}\right)^2 \]

4. **Simplify the right side**:
The right side simplifies to:
\[ A_2^2 = x^2 + y^2 \]
So we have:
\[ \left(\frac{x}{y} + \frac{y}{x}\right) A = x^2 + y^2 \]

5. **Express \(A\)**:
To find \(A\), we can rearrange the equation:
\[ A = \frac{x^2 + y^2}{\frac{x}{y} + \frac{y}{x}} \]

6. **Simplify the denominator**:
The denominator can be simplified:
\[ \frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy} \]
Thus, substituting this back into our expression for \(A\):
\[ A = \frac{x^2 + y^2}{\frac{x^2 + y^2}{xy}} = \frac{x^2 + y^2 \cdot xy}{x^2 + y^2} \]

7. **Final simplification**:
The \(x^2 + y^2\) terms cancel out:
\[ A = xy \]

### Conclusion
The third proportional to \(\frac{x}{y} + \frac{y}{x}\) and \(\sqrt{x^2 + y^2}\) is \(xy\).