If the A.M of two positive numbers a and b, `(a gt b)` is twice their G.M. , then a:b is :

Options

A. a)`2: sqrt3`
B. b)`2:7 + 4sqrt3`
C. c)`2+ sqrt3 : 2-sqrt3`
D. d)`7+4sqrt3 :7-4sqrt3`

Answer

To solve the problem, we need to find the ratio \( a:b \) given that the arithmetic mean (A.M) of two positive numbers \( a \) and \( b \) (where \( a > b \)) is twice their geometric mean (G.M).

### Step-by-Step Solution:

1. **Define A.M and G.M**:
- The arithmetic mean (A.M) of \( a \) and \( b \) is given by:
\[ A.M = \frac{a + b}{2} \]
- The geometric mean (G.M) of \( a \) and \( b \) is given by:
\[ G.M = \sqrt{ab} \]

2. **Set up the equation**:
- According to the problem, we have:
\[ A.M = 2 \times G.M \]
- Substituting the expressions for A.M and G.M, we get:
\[ \frac{a + b}{2} = 2 \sqrt{ab} \]

3. **Multiply both sides by 2**:
- To eliminate the fraction, multiply both sides by 2:
\[ a + b = 4 \sqrt{ab} \]

4. **Rearranging the equation**:
- Rearranging gives us:
\[ a + b - 4 \sqrt{ab} = 0 \]

5. **Using the identity for component and dividend**:
- We can express \( a \) and \( b \) in terms of their square roots:
Let \( x = \sqrt{a} \) and \( y = \sqrt{b} \). Then, we have:
\[ x^2 + y^2 = 4xy \]
- Rearranging gives:
\[ x^2 - 4xy + y^2 = 0 \]

6. **Factoring the quadratic**:
- This can be factored as:
\[ (x - 2y)^2 = 0 \]
- Thus, we find:
\[ x - 2y = 0 \quad \Rightarrow \quad x = 2y \]

7. **Substituting back to find the ratio**:
- Since \( x = \sqrt{a} \) and \( y = \sqrt{b} \), we have:
\[ \sqrt{a} = 2\sqrt{b} \]
- Squaring both sides gives:
\[ a = 4b \]

8. **Finding the ratio \( a:b \)**:
- The ratio \( a:b \) can be expressed as:
\[ \frac{a}{b} = \frac{4b}{b} = 4:1 \]

### Final Answer:
The ratio \( a:b \) is \( 4:1 \).