If the arithmetic mean of two numbers a and b,`a>b>0`, is five times their geometric mean, then `(a+b)/(a-b)` is equal to:

Options

A. `2+sqrt(3):2-sqrt(3)`
B. `7+4sqrt(3):7-4sqrt(3)`
C. `2:7+4sqrt(3)`
D. `2:sqrt(3)`

Explanation

To solve the problem step by step, we start with the given conditions and derive the required expression.

### Step 1: Understand the given conditions
We know that:
- The arithmetic mean (AM) of two numbers \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]
- The geometric mean (GM) of the same two numbers is given by:
\[ GM = \sqrt{ab} \]

### Step 2: Set up the equation based on the problem statement
According to the problem, the arithmetic mean is five times the geometric mean:
\[ \frac{a + b}{2} = 5 \sqrt{ab} \]

### Step 3: Multiply both sides by 2
To eliminate the fraction, we multiply both sides by 2:
\[ a + b = 10 \sqrt{ab} \]

### Step 4: Square both sides
Next, we square both sides to eliminate the square root:
\[ (a + b)^2 = (10 \sqrt{ab})^2 \]
This simplifies to:
\[ a^2 + 2ab + b^2 = 100ab \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ a^2 + b^2 + 2ab - 100ab = 0 \]
\[ a^2 + b^2 - 98ab = 0 \]

### Step 6: Use the identity for \( a - b \)
We can use the identity:
\[ a^2 - b^2 = (a + b)(a - b) \]
From the earlier step, we know:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting this in gives:
\[ (a + b)^2 - 2ab - 98ab = 0 \]
Thus:
\[ (a + b)^2 - 100ab = 0 \]

### Step 7: Express \( a - b \)
Now, we use the identity:
\[ a - b = \sqrt{(a + b)^2 - 4ab} \]
Substituting \( (a + b)^2 = 100ab \):
\[ a - b = \sqrt{100ab - 4ab} = \sqrt{96ab} \]

### Step 8: Find \( \frac{a + b}{a - b} \)
Now we need to find:
\[ \frac{a + b}{a - b} \]
Substituting the expressions we have:
\[ \frac{a + b}{a - b} = \frac{10 \sqrt{ab}}{\sqrt{96ab}} = \frac{10}{\sqrt{96}} = \frac{10}{4\sqrt{6}} = \frac{5}{2\sqrt{6}} \]

### Step 9: Rationalize the denominator
To rationalize the denominator:
\[ \frac{5}{2\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{5\sqrt{6}}{12} \]

### Final Answer
Thus, the value of \( \frac{a + b}{a - b} \) is:
\[ \frac{5\sqrt{6}}{12} \]