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 value of `C/A` is

Options

A. `-k/2`
B. `-k`
C. `k/2`
D. `k`

Explanation

To solve the problem step by step, we need to analyze the given information and apply the properties of polynomial roots.

### Step 1: Understand the Roots of the Polynomial
We have a polynomial equation \( x^4 + Ax^3 + Bx^2 + Cx + D = 0 \) with roots \( \alpha, \beta, \gamma, \delta \). According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta + \gamma + \delta = -A \)
- 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 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 product of the roots \( \alpha\beta\gamma\delta = D \)

### Step 2: Use the Given Condition
We are given that \( \alpha\beta = \gamma\delta = k \). This means:
- \( \alpha\beta = k \)
- \( \gamma\delta = k \)

### Step 3: Relate the Roots to the Coefficients
From Vieta's formulas, we can express \( C \) in terms of the roots:
\[ C = -(\alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta) \]
Substituting \( \alpha\beta = k \) and \( \gamma\delta = k \):
\[ C = -\left(k(\gamma + \delta) + k(\alpha + \beta)\right) \]

### Step 4: Simplify the Expression for C
We can rewrite the expression for \( C \):
\[ C = -k(\gamma + \delta + \alpha + \beta) \]

### Step 5: Find the Value of A
From the polynomial \( x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \), we need to find the roots \( A, B, C, D \) such that \( A + B = 0 \). This implies \( B = -A \).

### Step 6: Use Vieta's Formulas on the Second Polynomial
Using Vieta's formulas on the polynomial \( x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \):
- The sum of the roots \( A + B + C + D = 2 \)
- The sum of the products of the roots taken two at a time \( AB + AC + AD + BC + BD + CD = 4 \)
- The sum of the products of the roots taken three at a time \( ABC + ABD + ACD + BCD = -6 \)
- The product of the roots \( ABCD = -21 \)

### Step 7: Express C in Terms of A
From the earlier derived expression for \( C \):
\[ C = -k(\gamma + \delta + \alpha + \beta) \]
Since \( \alpha + \beta + \gamma + \delta = -A \), we can substitute:
\[ C = kA \]

### Step 8: Find the Ratio \( \frac{C}{A} \)
Now we can find the ratio:
\[ \frac{C}{A} = k \]

### Final Answer
Thus, the value of \( \frac{C}{A} \) is \( k \).

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