If `x,y in R` satisfy the equation `x^2 + y^2 - 4x-2y + 5 = 0,` then the value of the expression `[(sqrtx-sqrty)^2+4sqrt(xy)]/((x+sqrt(xy))` is

Explanation

To solve the problem step by step, we will start with the given equation and then evaluate the required expression.

### Step 1: Analyze the Given Equation
The given equation is:
\[ x^2 + y^2 - 4x - 2y + 5 = 0 \]

### Step 2: Rewrite the Equation
We can rewrite the equation by completing the square for both \(x\) and \(y\).

For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

For \(y\):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

Substituting these into the equation gives:
\[ (x - 2)^2 - 4 + (y - 1)^2 - 1 + 5 = 0 \]
\[ (x - 2)^2 + (y - 1)^2 = 0 \]

### Step 3: Solve for \(x\) and \(y\)
Since the sum of two squares is zero, both squares must be zero:
\[ (x - 2)^2 = 0 \quad \text{and} \quad (y - 1)^2 = 0 \]
Thus, we have:
\[ x - 2 = 0 \implies x = 2 \]
\[ y - 1 = 0 \implies y = 1 \]

### Step 4: Substitute \(x\) and \(y\) into the Expression
We need to evaluate the expression:
\[ \frac{(\sqrt{x} - \sqrt{y})^2 + 4\sqrt{xy}}{x + \sqrt{xy}} \]
Substituting \(x = 2\) and \(y = 1\):
\[ \frac{(\sqrt{2} - \sqrt{1})^2 + 4\sqrt{2 \cdot 1}}{2 + \sqrt{2 \cdot 1}} \]

### Step 5: Simplify the Expression
Calculating the numerator:
\[ (\sqrt{2} - 1)^2 + 4\sqrt{2} \]
Calculating \((\sqrt{2} - 1)^2\):
\[ (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]
Thus, the numerator becomes:
\[ 3 - 2\sqrt{2} + 4\sqrt{2} = 3 + 2\sqrt{2} \]

Calculating the denominator:
\[ 2 + \sqrt{2} \]

### Step 6: Final Expression
Now we have:
\[ \frac{3 + 2\sqrt{2}}{2 + \sqrt{2}} \]

### Step 7: Rationalize the Denominator
To simplify, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(3 + 2\sqrt{2})(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})} \]
Calculating the denominator:
\[ (2 + \sqrt{2})(2 - \sqrt{2}) = 4 - 2 = 2 \]
Calculating the numerator:
\[ (3 + 2\sqrt{2})(2 - \sqrt{2}) = 6 - 3\sqrt{2} + 4\sqrt{2} - 2 = 4 + \sqrt{2} \]

Thus, we have:
\[ \frac{4 + \sqrt{2}}{2} \]

### Step 8: Final Answer
This simplifies to:
\[ 2 + \frac{\sqrt{2}}{2} \]