The arithmetic mean of two positive numbers a and b exceeds their geometric mean by 2 and the harmonic mean is one - fifth of the greater of a and b, such that `alpha=a+b and beta=|a-b|`, then the value of `alpha+beta^(2)` is equal to

Options

A. 96
B. 234
C. 74
D. 84

Answer

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

### Step 1: Define the Means
The arithmetic mean (AM) of two numbers \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]
The geometric mean (GM) is given by:
\[ GM = \sqrt{ab} \]
According to the problem, the arithmetic mean exceeds the geometric mean by 2:
\[ \frac{a + b}{2} = \sqrt{ab} + 2 \]

### Step 2: Rearranging the Equation
Multiply both sides by 2 to eliminate the fraction:
\[ a + b = 2\sqrt{ab} + 4 \]
Rearranging gives:
\[ 2\sqrt{ab} = a + b - 4 \]

### Step 3: Square Both Sides
Square both sides to eliminate the square root:
\[ (2\sqrt{ab})^2 = (a + b - 4)^2 \]
This simplifies to:
\[ 4ab = (a + b)^2 - 8(a + b) + 16 \]

### Step 4: Expand and Rearrange
Expanding the right-hand side:
\[ 4ab = a^2 + 2ab + b^2 - 8a - 8b + 16 \]
Rearranging gives:
\[ 2ab - a^2 - b^2 + 8a + 8b - 16 = 0 \]

### Step 5: Use the Harmonic Mean Condition
The harmonic mean (HM) is given by:
\[ HM = \frac{2ab}{a + b} \]
According to the problem, the harmonic mean is one-fifth of the greater of \( a \) and \( b \). Assuming \( a \) is greater:
\[ \frac{2ab}{a + b} = \frac{a}{5} \]
Cross-multiplying gives:
\[ 10ab = a(a + b) \]
This simplifies to:
\[ 10b = a + b \quad \Rightarrow \quad 9b = a \quad \Rightarrow \quad a = 9b \]

### Step 6: Substitute into the Previous Equation
Substituting \( a = 9b \) into the equation from Step 4:
\[ 2(9b)b - (9b)^2 - b^2 + 8(9b) + 8b - 16 = 0 \]
This simplifies to:
\[ 18b^2 - 81b^2 - b^2 + 72b + 8b - 16 = 0 \]
Combining like terms:
\[ -64b^2 + 80b - 16 = 0 \]

### Step 7: Solve the Quadratic Equation
Dividing the entire equation by -16 gives:
\[ 4b^2 - 5b + 1 = 0 \]
Using the quadratic formula:
\[ b = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} = \frac{5 \pm \sqrt{25 - 16}}{8} = \frac{5 \pm 3}{8} \]
This gives us:
\[ b = 1 \quad \text{or} \quad b = \frac{1}{4} \]

### Step 8: Find Corresponding Values of \( a \)
Using \( a = 9b \):
1. If \( b = 1 \), then \( a = 9 \).
2. If \( b = \frac{1}{4} \), then \( a = \frac{9}{4} \).

### Step 9: Calculate \( \alpha \) and \( \beta \)
Now, we calculate \( \alpha = a + b \) and \( \beta = |a - b| \):
1. For \( a = 9 \) and \( b = 1 \):
\[ \alpha = 9 + 1 = 10, \quad \beta = |9 - 1| = 8 \]
2. For \( a = \frac{9}{4} \) and \( b = \frac{1}{4} \):
\[ \alpha = \frac{9}{4} + \frac{1}{4} = \frac{10}{4} = \frac{5}{2}, \quad \beta = |\frac{9}{4} - \frac{1}{4}| = \frac{8}{4} = 2 \]

### Step 10: Calculate \( \alpha + \beta^2 \)
Using the values from the first case:
\[ \alpha + \beta^2 = 10 + 8^2 = 10 + 64 = 74 \]

### Final Answer
Thus, the value of \( \alpha + \beta^2 \) is:
\[ \boxed{74} \]