If A, g and H are respectively arithmetic ,
geometric and harmomic means between a and b
both being unequal and positive, then
`A = (a + b)/(2) rArr a + b = 2A`
`G = sqrt(ab) rArr G^(2) = ab `
`H = (2ab)/(a+ b ) rArr G^(2) = AH`
On the basis of above information answer the following questions .
The sum of AM and GM of two positive numbers
equal to the difference between the numbers . the
numbers are in the ratio .

Options

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

Answer

To solve the problem, we need to find the ratio of two positive numbers \( A \) and \( B \) such that the sum of their Arithmetic Mean (AM) and Geometric Mean (GM) equals the difference between the two numbers.

Let's denote the two positive numbers as \( a \) and \( b \) where \( a > b \).

### Step 1: Write the expressions for AM and GM
The Arithmetic Mean (AM) of \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]

The Geometric Mean (GM) of \( a \) and \( b \) is given by:
\[ GM = \sqrt{ab} \]

### Step 2: Set up the equation based on the problem statement
According to the problem, the sum of AM and GM equals the difference between the numbers:
\[ AM + GM = a - b \]
Substituting the expressions for AM and GM, we get:
\[ \frac{a + b}{2} + \sqrt{ab} = a - b \]

### Step 3: Simplify the equation
Multiply the entire equation by 2 to eliminate the fraction:
\[ a + b + 2\sqrt{ab} = 2a - 2b \]
Rearranging gives:
\[ a + b + 2\sqrt{ab} + 2b = 2a \]
\[ a + 3b + 2\sqrt{ab} = 2a \]
Subtract \( a \) from both sides:
\[ 3b + 2\sqrt{ab} = a \]

### Step 4: Isolate \( a \)
From the equation \( a = 3b + 2\sqrt{ab} \), we can express \( a \) in terms of \( b \):
\[ a - 3b = 2\sqrt{ab} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (a - 3b)^2 = 4ab \]
Expanding the left side:
\[ a^2 - 6ab + 9b^2 = 4ab \]
Rearranging gives:
\[ a^2 - 10ab + 9b^2 = 0 \]

### Step 6: Factor the quadratic equation
This quadratic can be factored as:
\[ (a - 9b)(a - b) = 0 \]
Since \( a \) and \( b \) are unequal, we discard the solution \( a - b = 0 \) and take:
\[ a - 9b = 0 \implies a = 9b \]

### Step 7: Find the ratio \( \frac{a}{b} \)
From \( a = 9b \), we can find the ratio:
\[ \frac{a}{b} = \frac{9b}{b} = 9 \]
Thus, the ratio of \( a \) to \( b \) is:
\[ \text{Ratio } a : b = 9 : 1 \]

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

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