The roots `alpha, beta and gamma` of an equation `x^(3) - 3 a x^(2) + 3 bx - c = 0` are in H.P. Then,

Options

A. `beta = (1)/(a)`
B. `beta = b`
C. `beta = (b)/(c)`
D. `beta = (c)/(b)`

Answer

To solve the problem, we need to find the value of \( \beta \) given that the roots \( \alpha, \beta, \gamma \) of the cubic equation \( x^3 - 3ax^2 + 3bx - c = 0 \) are in Harmonic Progression (H.P.).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression (H.P.):**
If \( \alpha, \beta, \gamma \) are in H.P., then their reciprocals \( \frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma} \) are in Arithmetic Progression (A.P.). This means:
\[ \frac{1}{\beta} = \frac{1}{2} \left( \frac{1}{\alpha} + \frac{1}{\gamma} \right) \]

2. **Using Vieta's Formulas:**
From the given cubic equation \( x^3 - 3ax^2 + 3bx - c = 0 \), we can apply Vieta's formulas:
- Sum of the roots:
\[ \alpha + \beta + \gamma = 3a \]
- Sum of the product of the roots taken two at a time:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = 3b \]
- Product of the roots:
\[ \alpha\beta\gamma = c \]

3. **Expressing the A.P. Condition:**
From the H.P. condition, we can express:
\[ \frac{1}{\alpha} + \frac{1}{\gamma} = \frac{2}{\beta} \]
This implies:
\[ \frac{\alpha + \gamma}{\alpha\gamma} = \frac{2}{\beta} \]

4. **Finding \( \alpha + \gamma \):**
We can express \( \alpha + \gamma \) in terms of \( \beta \):
\[ \alpha + \gamma = 3a - \beta \]

5. **Finding \( \alpha\gamma \):**
Using the product of the roots:
\[ \alpha\beta\gamma = c \implies \alpha\gamma = \frac{c}{\beta} \]

6. **Substituting in the A.P. Condition:**
Substitute \( \alpha + \gamma \) and \( \alpha\gamma \) into the equation:
\[ \frac{3a - \beta}{\frac{c}{\beta}} = \frac{2}{\beta} \]
Simplifying gives:
\[ (3a - \beta)\beta = 2c \]
\[ 3a\beta - \beta^2 = 2c \]

7. **Rearranging the Equation:**
Rearranging gives us a quadratic equation in \( \beta \):
\[ \beta^2 - 3a\beta + 2c = 0 \]

8. **Using the Quadratic Formula:**
We can solve for \( \beta \) using the quadratic formula:
\[ \beta = \frac{-(-3a) \pm \sqrt{(-3a)^2 - 4 \cdot 1 \cdot 2c}}{2 \cdot 1} \]
\[ \beta = \frac{3a \pm \sqrt{9a^2 - 8c}}{2} \]

9. **Final Result for \( \beta \):**
The value of \( \beta \) can be expressed as:
\[ \beta = \frac{3a \pm \sqrt{9a^2 - 8c}}{2} \]