The harmonic mean of two positive numbers a and b is 4, their arithmetic mean is A and the geometric mean is G. If `2A+G^(2)=27, a+b=alpha` and `|a-b|=beta`, then the value of `(alpha)/(beta)` is equal to

Options

A. 1
B. 3
C. `(5)/(2)`
D. 5

Answer

To solve the problem step by step, we will use the definitions of harmonic mean, arithmetic mean, and geometric mean, along with the given equations.

### Step 1: Understand the means
The harmonic mean (HM) of two numbers \( a \) and \( b \) is given by:
\[ HM = \frac{2ab}{a+b} \]
We know that \( HM = 4 \), hence:
\[ \frac{2ab}{a+b} = 4 \]

### Step 2: Set up the equation
Let \( a + b = \alpha \) and \( ab = p \). Then we can rewrite the harmonic mean equation:
\[ \frac{2p}{\alpha} = 4 \]
Multiplying both sides by \( \alpha \):
\[ 2p = 4\alpha \quad \Rightarrow \quad p = 2\alpha \]

### Step 3: Find the arithmetic mean (A)
The arithmetic mean (A) is given by:
\[ A = \frac{a+b}{2} = \frac{\alpha}{2} \]

### Step 4: Find the geometric mean (G)
The geometric mean (G) is given by:
\[ G = \sqrt{ab} = \sqrt{p} = \sqrt{2\alpha} \]

### Step 5: Use the given equation
We have the equation:
\[ 2A + G^2 = 27 \]
Substituting the values of \( A \) and \( G \):
\[ 2\left(\frac{\alpha}{2}\right) + 2\alpha = 27 \]
This simplifies to:
\[ \alpha + 2\alpha = 27 \quad \Rightarrow \quad 3\alpha = 27 \quad \Rightarrow \quad \alpha = 9 \]

### Step 6: Find \( ab \) (p)
From the earlier step, we have:
\[ p = 2\alpha = 2 \times 9 = 18 \]

### Step 7: Calculate \( |a-b| \) (beta)
Using the formula:
\[ |a-b| = \sqrt{(a+b)^2 - 4ab} \]
Substituting \( a+b = \alpha = 9 \) and \( ab = p = 18 \):
\[ |a-b| = \sqrt{9^2 - 4 \times 18} = \sqrt{81 - 72} = \sqrt{9} = 3 \]
Thus, \( \beta = 3 \).

### Step 8: Calculate \( \frac{\alpha}{\beta} \)
Now we can find:
\[ \frac{\alpha}{\beta} = \frac{9}{3} = 3 \]

### Final Answer
The value of \( \frac{\alpha}{\beta} \) is \( 3 \).

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