Suppose `alpha, beta` are roots of `ax^(2)+bx+c=0` and `gamma, delta` are roots of `Ax^(2)+Bx+C=0`. If `a,b,c` are in GP as well as `alpha,beta, gamma, delta`, then `A,B,C` are in:

Options

A. (a) AP only
B. (b) GP only
C. (c) AP and GP
D. (d) None of these

Explanation

To solve the problem, we need to analyze the relationships between the roots and the coefficients of the quadratic equations given. Let's break this down step by step.

### Step 1: Understanding the Roots
Given two quadratic equations:
1. \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \)
2. \( Ax^2 + Bx + C = 0 \) with roots \( \gamma \) and \( \delta \)

From Vieta's formulas, we know:
- For the first equation:
- Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
- Product of roots: \( \alpha \beta = \frac{c}{a} \)

- For the second equation:
- Sum of roots: \( \gamma + \delta = -\frac{B}{A} \)
- Product of roots: \( \gamma \delta = \frac{C}{A} \)

### Step 2: Roots in Geometric Progression
We know that \( \alpha, \beta, \gamma, \delta \) are in geometric progression (GP). This implies:
\[ \frac{\beta}{\alpha} = \frac{\gamma}{\beta} = \frac{\delta}{\gamma} \]
Let’s denote the common ratio as \( r \). Therefore, we can express the roots as:
- \( \beta = \alpha r \)
- \( \gamma = \beta r = \alpha r^2 \)
- \( \delta = \gamma r = \alpha r^3 \)

### Step 3: Expressing the Sums and Products
Now we can express the sums and products of the roots in terms of \( \alpha \) and \( r \):
- Sum of roots \( \alpha + \beta = \alpha + \alpha r = \alpha(1 + r) \)
- Sum of roots \( \gamma + \delta = \alpha r^2 + \alpha r^3 = \alpha r^2(1 + r) \)

From Vieta’s formulas:
\[ -\frac{b}{a} = \alpha(1 + r) \quad \text{and} \quad -\frac{B}{A} = \alpha r^2(1 + r) \]

### Step 4: Setting Up the Ratios
Now, we can set up the ratio of the sums:
\[ \frac{\alpha(1 + r)}{\alpha r^2(1 + r)} = \frac{1}{r^2} \]
This simplifies to:
\[ \frac{1}{r^2} \]

Now for the products:
- Product of roots \( \alpha \beta = \alpha \cdot \alpha r = \alpha^2 r \)
- Product of roots \( \gamma \delta = \alpha r^2 \cdot \alpha r^3 = \alpha^2 r^5 \)

From Vieta’s formulas:
\[ \frac{c}{a} = \alpha^2 r \quad \text{and} \quad \frac{C}{A} = \alpha^2 r^5 \]

### Step 5: Setting Up the Product Ratios
Now we can set up the ratio of the products:
\[ \frac{\alpha^2 r}{\alpha^2 r^5} = \frac{1}{r^4} \]

### Step 6: Equating the Ratios
From the two ratios we derived:
1. \( \frac{\alpha + \beta}{\gamma + \delta} = \frac{1}{r^2} \)
2. \( \frac{\alpha \beta}{\gamma \delta} = \frac{1}{r^4} \)

We can square the first ratio:
\[ \left(\frac{1}{r^2}\right)^2 = \frac{1}{r^4} \]

This shows that the two expressions are equal.

### Conclusion
Since \( a, b, c \) are in GP, and we have shown that the relationships hold for \( A, B, C \) as well, we conclude that \( A, B, C \) are also in GP.

Thus, the final answer is:
**Option B: GP only.**