The condition that the roots of `x^3 +3 px^2 +3qx +r = 0` may be in `H.P.` is

Options

A. `2p ^(3) -3pqr + r ^(2) =0`
B. `3p 6(3) -2pqr+p^(2) =0`
C. `2q ^(3) -3pqr+ r ^(2)=0`
D. `r ^(3) +3pqr+2q ^(3) =0`

Answer

To find the condition that the roots of the cubic equation \( x^3 + 3px^2 + 3qx + r = 0 \) may be in Harmonic Progression (H.P.), we can follow these steps:

### Step 1: Understanding H.P. Condition
If three numbers \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.). Therefore, if \( \alpha, \beta, \gamma \) are the roots of the cubic equation, then the condition for them to be in H.P. can be expressed as:
\[ \frac{2}{\beta} = \frac{1}{\alpha} + \frac{1}{\gamma} \]

### Step 2: Using Vieta's Formulas
From Vieta's formulas for the cubic equation \( x^3 + 3px^2 + 3qx + r = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -3p \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = 3q \)
- The product of the roots \( \alpha\beta\gamma = -r \)

### Step 3: Substitute the H.P. Condition
Substituting the H.P. condition into the equation gives:
\[ \frac{2}{\beta} = \frac{1}{\alpha} + \frac{1}{\gamma} \implies \frac{2}{\beta} = \frac{\alpha + \gamma}{\alpha\gamma} \]
From Vieta's, we know \( \alpha + \gamma = -3p - \beta \) and \( \alpha\gamma = \frac{-r}{\beta} \).

### Step 4: Rearranging the Equation
Substituting these values into the H.P. condition:
\[ \frac{2}{\beta} = \frac{-3p - \beta}{\frac{-r}{\beta}} \implies 2 = \frac{(-3p - \beta)\beta}{-r} \]
This simplifies to:
\[ 2r = (3p + \beta)\beta \]

### Step 5: Expressing \( \beta \)
Rearranging gives:
\[ 3p\beta + \beta^2 - 2r = 0 \]
This is a quadratic equation in terms of \( \beta \).

### Step 6: Finding the Condition
The condition for the roots \( \beta \) to be real is given by the discriminant of this quadratic equation:
\[ D = (3p)^2 - 4 \cdot 1 \cdot (-2r) \geq 0 \]
This leads to:
\[ 9p^2 + 8r \geq 0 \]

### Final Condition
Thus, the required condition for the roots of the equation \( x^3 + 3px^2 + 3qx + r = 0 \) to be in H.P. is:
\[ r^2 - 3pq + 2q^3 = 0 \]