If `(a+b) : sqrt(ab) = 4:1`, where `a gt b gt 0`, then a: b is:

Options

A. `(2+ sqrt(3)): (2 - sqrt(3))`
B. `(2-sqrt(3)) : (2 + sqrt(3))`
C. `(3+ sqrt(2)) : (3-sqrt(2))`
D. `(3-sqrt(2)) : ( 3+ sqrt(2))`

Answer

To solve the problem `(a + b) : sqrt(ab) = 4 : 1` where `a > b > 0`, we need to find the ratio `a : b`.

### Step-by-step Solution:

1. **Set Up the Equation**:
Given the ratio, we can write:
\[ \frac{a + b}{\sqrt{ab}} = 4 \]
This implies:
\[ a + b = 4\sqrt{ab} \]

2. **Let `a` and `b` be expressed in terms of a single variable**:
Assume \( a = kb \) where \( k > 1 \) (since \( a > b \)). Then we can rewrite \( a + b \) and \( \sqrt{ab} \):
\[ a + b = kb + b = (k + 1)b \]
\[ \sqrt{ab} = \sqrt{kb \cdot b} = \sqrt{kb^2} = b\sqrt{k} \]

3. **Substitute into the equation**:
Substitute these expressions back into the equation:
\[ (k + 1)b = 4b\sqrt{k} \]

4. **Cancel `b` from both sides** (since \( b > 0 \)):
\[ k + 1 = 4\sqrt{k} \]

5. **Rearrange the equation**:
Rearranging gives:
\[ k + 1 - 4\sqrt{k} = 0 \]

6. **Let \( x = \sqrt{k} \)**:
Then \( k = x^2 \) and substituting gives:
\[ x^2 + 1 - 4x = 0 \]
This can be rearranged to:
\[ x^2 - 4x + 1 = 0 \]

7. **Use the quadratic formula**:
Applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]

8. **Find \( k \)**:
Since \( k = x^2 \), we have two possible values:
\[ k = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]
or
\[ k = (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

9. **Determine the ratio \( a : b \)**:
The ratio \( a : b \) can be expressed as:
\[ a : b = k : 1 \]
Therefore, we have two potential ratios:
\[ a : b = (7 + 4\sqrt{3}) : 1 \quad \text{or} \quad (7 - 4\sqrt{3}) : 1 \]

10. **Final Ratio**:
Since \( a > b \), we take the positive root:
\[ a : b = 2 + \sqrt{3} : 2 - \sqrt{3} \]

### Conclusion:
Thus, the final answer is:
\[ \boxed{2 + \sqrt{3} : 2 - \sqrt{3}} \]