If sum of the `A.M. & G.M.` of two positive distinct numbers is equal to the difference between the number then numbers are in ratio :

Options

A. `1:3`
B. `1:6`
C. `9:1`
D. `1:12`

Explanation

To solve the problem, we need to find the ratio of two distinct positive numbers \( a \) and \( b \) given that the sum of their Arithmetic Mean (A.M.) and Geometric Mean (G.M.) equals the difference between the two numbers.

### Step-by-step Solution:

1. **Define the Numbers:**
Let the two distinct positive numbers be \( a \) and \( b \).

2. **Calculate 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:
\[ G.M. = \sqrt{ab} \]

3. **Set Up the Equation:**
According to the problem, the sum of the A.M. and G.M. equals the difference between the numbers:
\[ \frac{a + b}{2} + \sqrt{ab} = a - b \]

4. **Eliminate the Fraction:**
To eliminate the fraction, multiply the entire equation by 2:
\[ a + b + 2\sqrt{ab} = 2(a - b) \]

5. **Rearrange the Equation:**
Rearranging gives:
\[ a + b + 2\sqrt{ab} = 2a - 2b \]
Simplifying further:
\[ 2\sqrt{ab} = 2a - a - 3b \]
This simplifies to:
\[ 2\sqrt{ab} = a - 3b \]

6. **Square Both Sides:**
Squaring both sides to eliminate the square root:
\[ (2\sqrt{ab})^2 = (a - 3b)^2 \]
This results in:
\[ 4ab = a^2 - 6ab + 9b^2 \]

7. **Rearranging the Equation:**
Rearranging gives:
\[ 4ab + 6ab - 9b^2 - a^2 = 0 \]
Which simplifies to:
\[ a^2 - 10ab + 9b^2 = 0 \]

8. **Factoring the Quadratic:**
This quadratic can be factored as:
\[ (a - 9b)(a - b) = 0 \]

9. **Finding the Cases:**
From the factored form, we have two cases:
- Case 1: \( a - b = 0 \) (not possible since \( a \) and \( b \) are distinct)
- Case 2: \( a - 9b = 0 \) which gives \( a = 9b \)

10. **Finding the Ratio:**
Therefore, the ratio of \( a \) to \( b \) is:
\[ \frac{a}{b} = \frac{9b}{b} = 9 \]
This means the numbers are in the ratio:
\[ 9:1 \]

### Final Answer:
The numbers are in the ratio \( 9:1 \).