Consider two positive numbers a and b. If arithmetic mean of a and b exceeds their geometric mean by 3/2, and geometric mean of aand b exceeds their harmonic mean by 6/5 then the value of `a^2+b^2` will be

Options

A. 150
B. 153
C. 156
D. 159

Answer

To solve the problem step by step, we will follow the given information about the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) of two positive numbers \( a \) and \( b \).

### Step 1: Define the Means
1. **Arithmetic Mean (AM)**:
\[ AM = \frac{a + b}{2} \]

2. **Geometric Mean (GM)**:
\[ GM = \sqrt{ab} \]

3. **Harmonic Mean (HM)**:
\[ HM = \frac{2ab}{a + b} \]

### Step 2: Set Up the Equations
From the problem statement, we have two conditions:
1. The arithmetic mean exceeds the geometric mean by \( \frac{3}{2} \):
\[ \frac{a + b}{2} = \sqrt{ab} + \frac{3}{2} \]

2. The geometric mean exceeds the harmonic mean by \( \frac{6}{5} \):
\[ \sqrt{ab} = \frac{2ab}{a + b} + \frac{6}{5} \]

### Step 3: Rearranging the First Equation
From the first equation:
\[ \frac{a + b}{2} - \sqrt{ab} = \frac{3}{2} \]
Multiply through by 2:
\[ a + b - 2\sqrt{ab} = 3 \]
Rearranging gives:
\[ a + b = 2\sqrt{ab} + 3 \quad \text{(Equation 1)} \]

### Step 4: Rearranging the Second Equation
From the second equation:
\[ \sqrt{ab} - \frac{2ab}{a + b} = \frac{6}{5} \]
Multiply through by \( 5(a + b) \):
\[ 5\sqrt{ab}(a + b) - 10ab = 6(a + b) \]
Rearranging gives:
\[ 5\sqrt{ab}(a + b) - 6(a + b) = 10ab \]
Factoring out \( (a + b) \):
\[ (a + b)(5\sqrt{ab} - 6) = 10ab \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
Substituting \( a + b = 2\sqrt{ab} + 3 \) into Equation 2:
\[ (2\sqrt{ab} + 3)(5\sqrt{ab} - 6) = 10ab \]
Expanding gives:
\[ 10ab + 15\sqrt{ab} - 12\sqrt{ab} - 18 = 10ab \]
Simplifying:
\[ 3\sqrt{ab} - 18 = 0 \]
Thus:
\[ \sqrt{ab} = 6 \implies ab = 36 \]

### Step 6: Find \( a + b \)
Substituting \( ab = 36 \) back into Equation 1:
\[ a + b = 2\sqrt{36} + 3 = 12 + 3 = 15 \quad \text{(Equation 3)} \]

### Step 7: Use the Identity to Find \( a^2 + b^2 \)
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ 15^2 = a^2 + b^2 + 2 \cdot 36 \]
Calculating:
\[ 225 = a^2 + b^2 + 72 \]
Thus:
\[ a^2 + b^2 = 225 - 72 = 153 \]

### Final Answer
The value of \( a^2 + b^2 \) is:
\[ \boxed{153} \]