If `alpha, beta, gamma` are the roots of the equation `x^(4)+Ax^(3)+Bx^(2)+Cx+D=0` such that `alpha beta= gamma delta=k` and A,B,C,D are the roots of `x^(4)-2x^(3)+4x^(2)+6x-21=0` such that `A+B=0`
The correct statement is

Options

A. `C^(2)=AD`
B. `C^(2)=A^(2)D`
C. `C^(2)=AD^(2)`
D. `C^(2)=(AD)^(2)`

Explanation

To solve the problem step by step, we will analyze the given information and derive the necessary relationships.

### Step 1: Understand the roots of the equations
We have two sets of roots:
1. The roots of the polynomial \( x^4 + Ax^3 + Bx^2 + Cx + D = 0 \) are \( \alpha, \beta, \gamma, \delta \).
2. The roots of the polynomial \( x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \) are \( A, B, C, D \).

### Step 2: Use Vieta's formulas for the first polynomial
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta + \gamma + \delta = -A \).
- The sum of the products of the roots taken two at a time \( \alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = B \).
- The sum of the products of the roots taken three at a time \( \alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = -C \).
- The product of the roots \( \alpha\beta\gamma\delta = D \).

### Step 3: Given relationships
We are given that:
- \( \alpha\beta = \gamma\delta = k \).
- Therefore, we can express the product of all roots as:
\[ \alpha\beta\gamma\delta = k \cdot \gamma\delta = k^2 = D. \]

### Step 4: Analyze the second polynomial
For the polynomial \( x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \):
- The sum of the roots \( A + B + C + D = 2 \).
- We are also given that \( A + B = 0 \), which implies \( B = -A \).

### Step 5: Substitute and derive relationships
From \( A + B + C + D = 2 \) and \( B = -A \):
\[ A - A + C + D = 2 \implies C + D = 2. \]
Using \( D = k^2 \), we can substitute:
\[ C + k^2 = 2 \implies C = 2 - k^2. \]

### Step 6: Relate \( C \) and \( A \)
From the relationship \( k = \frac{C}{A} \):
\[ k^2 = \left(\frac{C}{A}\right)^2 = \frac{C^2}{A^2}. \]
Substituting \( C = 2 - k^2 \) into the equation gives:
\[ k^2 = \frac{(2 - k^2)^2}{A^2}. \]

### Step 7: Solve for \( C^2 \)
Squaring both sides:
\[ k^2 A^2 = (2 - k^2)^2. \]
Expanding the right side:
\[ k^2 A^2 = 4 - 4k^2 + k^4. \]
Rearranging gives:
\[ k^4 - k^2 A^2 - 4 + 4k^2 = 0 \implies k^4 + (4 - A^2)k^2 - 4 = 0. \]

### Step 8: Identify the correct statement
From the derived relationships, we can conclude:
\[ C^2 = A^2 D. \]
Thus, the correct statement is:
\[ C^2 = A^2 D. \]

### Final Answer
The correct statement is \( C^2 = A^2 D \).