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, we will follow these steps systematically:

### Step 1: Use the definition of the harmonic mean
The harmonic mean \( H \) of two numbers \( a \) and \( b \) is given by the formula:
\[ H = \frac{2ab}{a + b} \]
Given that the harmonic mean is 4, we can set up the equation:
\[ \frac{2ab}{a + b} = 4 \]
Multiplying both sides by \( a + b \) gives:
\[ 2ab = 4(a + b) \]

### Step 2: Rearranging the equation
Rearranging the equation from Step 1, we have:
\[ 2ab = 4a + 4b \]
Dividing through by 2:
\[ ab = 2a + 2b \]

### Step 3: Use the definition of the arithmetic mean
The arithmetic mean \( A \) of \( a \) and \( b \) is:
\[ A = \frac{a + b}{2} \]
Thus, we can express \( a + b \) as:
\[ a + b = 2A \]

### Step 4: Use the definition of the geometric mean
The geometric mean \( G \) is given by:
\[ G = \sqrt{ab} \]
Thus, \( G^2 = ab \).

### Step 5: Substitute into the equation
We know from the problem statement that:
\[ 2A + G^2 = 27 \]
Substituting \( G^2 = ab \) into this equation gives:
\[ 2A + ab = 27 \]

### Step 6: Substitute \( ab \) from Step 2
From Step 2, we know that \( ab = 2a + 2b \). Substituting \( a + b = 2A \) into this gives:
\[ ab = 2(2A) = 4A \]
Now substituting this into the equation from Step 5:
\[ 2A + 4A = 27 \]
This simplifies to:
\[ 6A = 27 \implies A = \frac{27}{6} = 4.5 \]

### Step 7: Find \( a + b \) (denote as \( \alpha \))
Using \( a + b = 2A \):
\[ a + b = 2 \times 4.5 = 9 \]
Thus, \( \alpha = 9 \).

### Step 8: Find \( ab \) (denote as \( \beta \))
From \( ab = 4A \):
\[ ab = 4 \times 4.5 = 18 \]

### Step 9: Set up the quadratic equation
We have \( a + b = 9 \) and \( ab = 18 \). The numbers \( a \) and \( b \) are the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \]
Substituting the values we found:
\[ x^2 - 9x + 18 = 0 \]

### Step 10: Solve the quadratic equation
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -9, c = 18 \):
\[ x = \frac{9 \pm \sqrt{81 - 72}}{2} = \frac{9 \pm 3}{2} \]
This gives us:
\[ x = \frac{12}{2} = 6 \quad \text{or} \quad x = \frac{6}{2} = 3 \]
Thus, \( a = 6 \) and \( b = 3 \) (or vice versa).

### Step 11: Calculate \( |a - b| \) (denote as \( \beta \))
Now, we find \( |a - b| \):
\[ |a - b| = |6 - 3| = 3 \]
Thus, \( \beta = 3 \).

### Step 12: Calculate \( \frac{\alpha}{\beta} \)
Finally, we calculate:
\[ \frac{\alpha}{\beta} = \frac{9}{3} = 3 \]

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

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