If A, G, H are respectively the A.M., G.M., H.M. of three numhers `alpha, beta, gamma`, then the cquation whose roots are `alpha, beta, gamma` is

Options

A. `x^(3)-3Ax^(2)+(G^(3))/(H)x+G^(3)=0`
B. `x^(3)-3Ax^(2)+(G^(3))/(H)x-G^(3)=0`
C. `x^(3)-3Ax^(2)-(G^(3))/(H)x-G^(3)=0`
D. none of these

Answer

To find the equation whose roots are the three numbers \( \alpha, \beta, \gamma \) given that \( A, G, H \) are their Arithmetic Mean, Geometric Mean, and Harmonic Mean respectively, we can follow these steps:

### Step 1: Understand the definitions
1. **Arithmetic Mean (A)**:
\[ A = \frac{\alpha + \beta + \gamma}{3} \]
Therefore,
\[ \alpha + \beta + \gamma = 3A \]

2. **Geometric Mean (G)**:
\[ G = \sqrt[3]{\alpha \beta \gamma} \]
Thus,
\[ \alpha \beta \gamma = G^3 \]

3. **Harmonic Mean (H)**:
\[ H = \frac{3}{\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}} = \frac{3 \alpha \beta \gamma}{\alpha \beta + \beta \gamma + \gamma \alpha} \]
Rearranging gives:
\[ \alpha \beta + \beta \gamma + \gamma \alpha = \frac{3 \alpha \beta \gamma}{H} \]

### Step 2: Formulate the cubic equation
The general form of a cubic equation with roots \( \alpha, \beta, \gamma \) is:
\[ x^3 - Sx^2 + SPx - P = 0 \]
where:
- \( S = \alpha + \beta + \gamma = 3A \)
- \( P = \alpha \beta \gamma = G^3 \)
- \( SP = \alpha \beta + \beta \gamma + \gamma \alpha = \frac{3G^3}{H} \)

### Step 3: Substitute values into the cubic equation
Substituting the values we found into the cubic equation:
\[ x^3 - (3A)x^2 + \left(\frac{3G^3}{H}\right)x - G^3 = 0 \]

### Final Equation
Thus, the required cubic equation whose roots are \( \alpha, \beta, \gamma \) is:
\[ x^3 - 3Ax^2 + \frac{3G^3}{H}x - G^3 = 0 \]