If the AM of two positive numbers a and b `(agtb)` is twice of their GM, then `a:b` is

Options

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

Answer

To solve the problem, we need to establish the relationship between the arithmetic mean (AM) and the geometric mean (GM) of the two positive numbers \( a \) and \( b \).

### Step-by-Step Solution:

1. **Understanding the Means**:
- The arithmetic mean (AM) of two numbers \( a \) and \( b \) is given by:
\[ \text{AM} = \frac{a + b}{2} \]
- The geometric mean (GM) of two numbers \( a \) and \( b \) is given by:
\[ \text{GM} = \sqrt{ab} \]

2. **Setting Up the Equation**:
- According to the problem, the AM is twice the GM:
\[ \frac{a + b}{2} = 2\sqrt{ab} \]

3. **Multiplying 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. **Applying the AM-GM Inequality**:
- We can apply the AM-GM inequality, which states that for any two positive numbers:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
- However, we already know from our equation that:
\[ a + b = 4\sqrt{ab} \]
- This implies that equality holds in the AM-GM inequality, which occurs when \( a = b \).

6. **Setting \( a \) in terms of \( b \)**:
- Let \( a = kb \) for some positive constant \( k \). Then substituting into the equation:
\[ kb + b = 4\sqrt{kb \cdot b} \]
- Simplifying gives:
\[ (k + 1)b = 4\sqrt{kb^2} \]
- This simplifies to:
\[ (k + 1)b = 4b\sqrt{k} \]

7. **Dividing by \( b \)** (since \( b \neq 0 \)):
- We can divide both sides by \( b \):
\[ k + 1 = 4\sqrt{k} \]

8. **Squaring Both Sides**:
- Squaring both sides gives:
\[ (k + 1)^2 = 16k \]
- Expanding the left side:
\[ k^2 + 2k + 1 = 16k \]
- Rearranging gives:
\[ k^2 - 14k + 1 = 0 \]

9. **Using the Quadratic Formula**:
- We can solve for \( k \) using the quadratic formula:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{14 \pm \sqrt{196 - 4}}{2} = \frac{14 \pm \sqrt{192}}{2} \]
- Simplifying further:
\[ k = \frac{14 \pm 8\sqrt{3}}{2} = 7 \pm 4\sqrt{3} \]

10. **Finding the Ratio \( a:b \)**:
- Therefore, the ratio \( a:b \) can be expressed as:
\[ a:b = k:1 = (7 \pm 4\sqrt{3}):1 \]

### Final Answer:
The ratio \( a:b \) is \( 7 + 4\sqrt{3} : 1 \) or \( 7 - 4\sqrt{3} : 1 \).