If the roots of the equation `x^3 +3px^2 + 3qx -8=0` are in a geometric progression , then ` (q^3)/( p^3)`=

Options

A. 1
B. `-2`
C. `4`
D. `-8`

Answer

To solve the problem, we need to find the value of \(\frac{q^3}{p^3}\) given that the roots of the equation \(x^3 + 3px^2 + 3qx - 8 = 0\) are in geometric progression.

### Step-by-Step Solution:

1. **Identify the Roots**:
Since the roots are in geometric progression, we can denote them as:
\[ \alpha, \quad \alpha r, \quad \alpha r^2 \]

2. **Use Vieta's Formulas**:
According to Vieta's formulas for a cubic equation \(x^3 + ax^2 + bx + c = 0\), we have:
- Sum of the roots: \(\alpha + \alpha r + \alpha r^2 = -3p\)
- Sum of the product of the roots taken two at a time: \(\alpha \cdot \alpha r + \alpha \cdot \alpha r^2 + \alpha r \cdot \alpha r^2 = 3q\)
- Product of the roots: \(\alpha \cdot \alpha r \cdot \alpha r^2 = -(-8) = 8\)

3. **Calculate the Product of the Roots**:
The product of the roots can be simplified:
\[ \alpha^3 r^3 = 8 \]
Taking the cube root gives:
\[ \alpha r = 2 \]

4. **Substituting a Root**:
We can substitute \(x = 2\) into the original equation to find a relationship between \(p\) and \(q\):
\[ 2^3 + 3p(2^2) + 3q(2) - 8 = 0 \]
Simplifying this:
\[ 8 + 12p + 6q - 8 = 0 \]
This simplifies to:
\[ 12p + 6q = 0 \]
Rearranging gives:
\[ 2p + q = 0 \quad \Rightarrow \quad q = -2p \]

5. **Finding \(\frac{q}{p}\)**:
From \(q = -2p\), we can express \(\frac{q}{p}\):
\[ \frac{q}{p} = -2 \]

6. **Finding \(\frac{q^3}{p^3}\)**:
Now, we cube both sides:
\[ \left(\frac{q}{p}\right)^3 = (-2)^3 = -8 \]
Thus, we have:
\[ \frac{q^3}{p^3} = -8 \]

### Final Answer:
\[ \frac{q^3}{p^3} = -8 \]