If `x+sqrt(5)=5+sqrt(y)` and `x,y` are positive integers , then the value of `(sqrt(x)+y)/(x+sqrt(y))` is

Options

A. `1`
B. `2`
C. `sqrt(5)`
D. `5`

Answer

To solve the equation \( x + \sqrt{5} = 5 + \sqrt{y} \) where \( x \) and \( y \) are positive integers, we will follow these steps:

### Step 1: Separate the rational and irrational parts
We start with the equation:
\[ x + \sqrt{5} = 5 + \sqrt{y} \]
We can separate the rational and irrational parts. The rational parts are \( x \) and \( 5 \), and the irrational parts are \( \sqrt{5} \) and \( \sqrt{y} \).

### Step 2: Set the rational parts equal
From the rational parts, we have:
\[ x = 5 \]

### Step 3: Set the irrational parts equal
From the irrational parts, we have:
\[ \sqrt{5} = \sqrt{y} \]
Squaring both sides gives us:
\[ 5 = y \]

### Step 4: Substitute the values of \( x \) and \( y \)
Now we have found that:
\[ x = 5 \quad \text{and} \quad y = 5 \]

### Step 5: Calculate the expression \( \frac{\sqrt{x} + y}{x + \sqrt{y}} \)
We need to evaluate:
\[ \frac{\sqrt{x} + y}{x + \sqrt{y}} \]
Substituting the values of \( x \) and \( y \):
\[ \frac{\sqrt{5} + 5}{5 + \sqrt{5}} \]

### Step 6: Simplify the expression
The expression can be simplified as follows:
\[ \frac{\sqrt{5} + 5}{5 + \sqrt{5}} = 1 \]

Thus, the value of \( \frac{\sqrt{x} + y}{x + \sqrt{y}} \) is:
\[ \boxed{1} \]