If the roots of the equation `x^3+3ax^2+3bx+c=0` are in `H.P.`, then (i) `2b^2=c(3ab-c)` (ii) `2b^3=c(3ab-c)` (iii) `2b^3=c^(2)(3ab-c)` (iv) `2b^2=c^(2)(3ab-c)`

Options

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

Explanation

To solve the problem, we need to analyze the given cubic equation \( x^3 + 3ax^2 + 3bx + c = 0 \) and determine the conditions under which its roots are in Harmonic Progression (H.P.).

### Step-by-Step Solution:

1. **Understanding Roots in H.P.**:
If the roots \( \alpha, \beta, \gamma \) are in H.P., then their reciprocals \( \frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma} \) are in Arithmetic Progression (A.P.). This means:
\[ 2 \cdot \frac{1}{\beta} = \frac{1}{\alpha} + \frac{1}{\gamma} \]
Rearranging gives:
\[ 2 = \frac{\alpha + \gamma}{\beta} \]
Thus, we can express \( \alpha + \gamma = 2 \cdot \beta \).

2. **Using Vieta's Formulas**:
From Vieta's formulas for the cubic equation \( x^3 + 3ax^2 + 3bx + c = 0 \):
- The sum of the roots:
\[ \alpha + \beta + \gamma = -3a \]
- The sum of the products of the roots taken two at a time:
\[ \alpha \beta + \beta \gamma + \gamma \alpha = 3b \]
- The product of the roots:
\[ \alpha \beta \gamma = -c \]

3. **Substituting for \( \alpha + \gamma \)**:
From the first step, we have \( \alpha + \gamma = 2\beta \). Substituting this into the sum of the roots:
\[ 2\beta + \beta = -3a \implies 3\beta = -3a \implies \beta = -a \]

4. **Finding \( \alpha + \gamma \)**:
Now substituting \( \beta = -a \) back into \( \alpha + \gamma \):
\[ \alpha + \gamma = 2(-a) = -2a \]

5. **Finding \( \alpha \gamma \)**:
Using the second Vieta's relation:
\[ \alpha(-a) + (-a)\gamma + \alpha\gamma = 3b \]
This simplifies to:
\[ -a(\alpha + \gamma) + \alpha\gamma = 3b \]
Substituting \( \alpha + \gamma = -2a \):
\[ -a(-2a) + \alpha\gamma = 3b \implies 2a^2 + \alpha\gamma = 3b \implies \alpha\gamma = 3b - 2a^2 \]

6. **Using the product of roots**:
From Vieta's, we also know:
\[ \alpha \beta \gamma = -c \implies \alpha(-a)\gamma = -c \implies -a\alpha\gamma = -c \implies a\alpha\gamma = c \]
Substituting \( \alpha\gamma = 3b - 2a^2 \):
\[ a(3b - 2a^2) = c \]

7. **Rearranging the equation**:
Rearranging gives:
\[ 3ab - 2a^3 = c \]
Thus, we can express \( c \) in terms of \( a \) and \( b \).

8. **Finding the relationships**:
To find the relationships between \( b \) and \( c \), we can manipulate the equation:
\[ 2b^2 = c(3ab - c) \]
This leads us to conclude that:
\[ 2b^2 = c(3ab - c) \]
which matches option (i).

### Conclusion:
The correct option is:
**(i) \( 2b^2 = c(3ab - c) \)**.