IF the roots of ` ax^3 +bx^2 +cx +d =0` are in G.P then the roots of `ay^3 +bky^2 +ck^2 y+ dk^3=0` are in

Options

A. A.P
B. G.P
C. H.P
D. A.G.P

Answer

To solve the problem, we need to determine the nature of the roots of the polynomial \( ay^3 + bky^2 + ck^2y + dk^3 = 0 \) given that the roots of the polynomial \( ax^3 + bx^2 + cx + d = 0 \) are in geometric progression (G.P.).

### Step-by-Step Solution:

1. **Identify the Roots of the First Polynomial:**
Let the roots of the polynomial \( ax^3 + bx^2 + cx + d = 0 \) be \( \alpha, \beta, \gamma \). Since these roots are in G.P., we can express them as:
\[ \beta = \alpha r \quad \text{and} \quad \gamma = \alpha r^2 \]
for some common ratio \( r \).

2. **Substituting Roots into the First Polynomial:**
The roots \( \alpha, \beta, \gamma \) satisfy the equation:
\[ a\alpha^3 + b\alpha^2 + c\alpha + d = 0 \]
Similarly, we can write the equations for \( \beta \) and \( \gamma \):
\[ a(\alpha r)^3 + b(\alpha r)^2 + c(\alpha r) + d = 0 \]
\[ a(\alpha r^2)^3 + b(\alpha r^2)^2 + c(\alpha r^2) + d = 0 \]

3. **Finding the Roots of the Second Polynomial:**
Now, consider the second polynomial \( ay^3 + bky^2 + ck^2y + dk^3 = 0 \). We will substitute \( y = kx \) into this polynomial:
\[ a(kx)^3 + b(kx)^2 + c(kx) + d = 0 \]
Simplifying this gives:
\[ ak^3x^3 + bk^2x^2 + ckx + d = 0 \]

4. **Factoring Out \( k^3 \):**
We can factor out \( k^3 \) from the first term:
\[ k^3 \left( a x^3 + \frac{b}{k} x^2 + \frac{c}{k^2} x + \frac{d}{k^3} \right) = 0 \]
Since \( k^3 \neq 0 \), we focus on the polynomial:
\[ a x^3 + \frac{b}{k} x^2 + \frac{c}{k^2} x + \frac{d}{k^3} = 0 \]

5. **Relating the Roots:**
The roots of this polynomial are \( \alpha, \beta, \gamma \) scaled by \( k \):
\[ k\alpha, k\beta, k\gamma \]
Since \( \alpha, \beta, \gamma \) are in G.P., multiplying them by a constant \( k \) keeps them in G.P. (as the ratio remains the same).

6. **Conclusion:**
Therefore, the roots of the polynomial \( ay^3 + bky^2 + ck^2y + dk^3 = 0 \) are also in G.P.

### Final Answer:
The roots of the polynomial \( ay^3 + bky^2 + ck^2y + dk^3 = 0 \) are in **G.P.**.