The third proportional to
`(x/y + y/x) andsqrt(x^2 + y^2)` is

Options

A. xy
B. `sqrt(xy)`
C. `root3(xy)`
D. `root4 (xy)`

Answer

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

### Step 1: Identify the expressions
Let:
- \( a = \frac{x}{y} + \frac{y}{x} \)
- \( b = \sqrt{x^2 + y^2} \)

### Step 2: Use the definition of third proportional
The third proportional \( c \) to \( a \) and \( b \) is defined such that:
\[ \frac{a}{b} = \frac{b}{c} \]
From this relationship, we can rearrange to find \( c \):
\[ c = \frac{b^2}{a} \]

### Step 3: Calculate \( a \)
First, we calculate \( a \):
\[ a = \frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy} \]

### Step 4: Calculate \( b \)
Next, we calculate \( b \):
\[ b = \sqrt{x^2 + y^2} \]

### Step 5: Substitute \( a \) and \( b \) into the formula for \( c \)
Now, substitute \( a \) and \( b \) into the formula for \( c \):
\[ c = \frac{b^2}{a} = \frac{(\sqrt{x^2 + y^2})^2}{\frac{x^2 + y^2}{xy}} = \frac{x^2 + y^2}{\frac{x^2 + y^2}{xy}} \]

### Step 6: Simplify the expression for \( c \)
This simplifies to:
\[ c = \frac{x^2 + y^2}{1} \cdot \frac{xy}{x^2 + y^2} = xy \]

### Conclusion
Thus, the third proportional to \( \left( \frac{x}{y} + \frac{y}{x} \right) \) and \( \sqrt{x^2 + y^2} \) is:
\[ \boxed{xy} \]