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

Options

A. A)xy
B. B)`sqrt(xy)`
C. C)`3 sqrt(xy)`
D. D)` 4 sqrt( xy)`

Answer

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

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

### Step 2: Find the third proportional
The third proportional \( c \) to the numbers \( a \) and \( b \) can be found using the formula:
\[ c = \frac{b^2}{a} \]

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

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

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

### Step 6: Simplify \( c \)
The \( x^2 + y^2 \) terms cancel out:
\[ c = xy \]

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