If `a`, `b`, `c` are in geometric progresion and the roots of the equations `ax^(2)+2bx+c=0` are `alpha` and `beta` and those of `cx^(2)+2bx+a=0` are `gamma` and `delta` then

Options

A. `alpha ne beta ne gamma ne delta`
B. `alpha ne beta` and `gamma ne delta`
C. `a alpha = a beta = c gamma = c delta`
D. `alpha=beta` , `gamma ne delta`

Explanation

To solve the problem, we will analyze the given equations and the conditions provided.

### Step 1: Understand the given conditions
We know that \( a, b, c \) are in geometric progression. This means that:
\[ \frac{b^2}{ac} = 1 \quad \text{(1)} \]

### Step 2: Analyze the roots of the first equation
The first equation is:
\[ ax^2 + 2bx + c = 0 \]
Let the roots of this equation be \( \alpha \) and \( \beta \). By Vieta's formulas, we have:
\[ \alpha + \beta = -\frac{2b}{a} \quad \text{(2)} \]
\[ \alpha \beta = \frac{c}{a} \quad \text{(3)} \]

### Step 3: Analyze the roots of the second equation
The second equation is:
\[ cx^2 + 2bx + a = 0 \]
Let the roots of this equation be \( \gamma \) and \( \delta \). Again, by Vieta's formulas, we have:
\[ \gamma + \delta = -\frac{2b}{c} \quad \text{(4)} \]
\[ \gamma \delta = \frac{a}{c} \quad \text{(5)} \]

### Step 4: Relate the roots of both equations
Since \( a, b, c \) are in geometric progression, we can express \( b \) in terms of \( a \) and \( c \):
\[ b = \sqrt{ac} \quad \text{(6)} \]

### Step 5: Substitute \( b \) into the equations
Substituting equation (6) into equations (2) and (4):
From (2):
\[ \alpha + \beta = -\frac{2\sqrt{ac}}{a} = -\frac{2\sqrt{c}}{\sqrt{a}} \quad \text{(7)} \]
From (4):
\[ \gamma + \delta = -\frac{2\sqrt{ac}}{c} = -\frac{2\sqrt{a}}{\sqrt{c}} \quad \text{(8)} \]

### Step 6: Analyze the product of the roots
From equations (3) and (5):
Using (3):
\[ \alpha \beta = \frac{c}{a} \quad \text{(9)} \]
Using (5):
\[ \gamma \delta = \frac{a}{c} \quad \text{(10)} \]

### Step 7: Establish relationships between roots
From the properties of the equations, we can establish that:
- The roots of the second equation are the reciprocals of the roots of the first equation, i.e., \( \gamma = \frac{1}{\alpha} \) and \( \delta = \frac{1}{\beta} \).

### Step 8: Conclusion
Since \( \alpha \) and \( \beta \) are roots of the first equation and \( \gamma \) and \( \delta \) are their reciprocals, we can conclude:
\[ \alpha \beta = \frac{c}{a}, \quad \gamma \delta = \frac{a}{c} \]
This implies:
\[ \alpha = \beta \quad \text{and} \quad \gamma = \delta \quad \text{(since they are reciprocals)} \]

Thus, the correct answer is:
**Option C: \( \alpha = \beta \) and \( \gamma = \delta \)**.