If `alpha, beta` are the roots of ` a x^2 + bx + c = 0` and `k in R` then the condition so that `alpha < k < beta` is :

Options

A. `ak^(2) + bk + c lt 0`
B. `a^(2) k^(2)+ abk + ac lt 0`
C. `a^(2)k^(2) + abk + ac gt 0`
D. none of these

Explanation

To solve the problem, we need to find the condition under which \( k \) lies between the roots \( \alpha \) and \( \beta \) of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

1. **Understanding the Roots**:
The roots \( \alpha \) and \( \beta \) of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( \alpha \) and \( \beta \) are real if the discriminant \( b^2 - 4ac \geq 0 \).

2. **Condition for \( k \) to be between \( \alpha \) and \( \beta \)**:
For \( k \) to lie between \( \alpha \) and \( \beta \), we need to check the sign of the quadratic function \( f(k) = ak^2 + bk + c \) evaluated at \( k \).

3. **Evaluating \( f(k) \)**:
The condition for \( k \) to be between \( \alpha \) and \( \beta \) can be expressed as:
\[ a \cdot f(k) < 0 \]
This means:
\[ a(ak^2 + bk + c) < 0 \]

4. **Expanding the Expression**:
Expanding the expression gives:
\[ a^2k^2 + abk + ac < 0 \]

5. **Conclusion**:
Therefore, the condition for \( k \) to lie between the roots \( \alpha \) and \( \beta \) is:
\[ a^2k^2 + abk + ac < 0 \]

### Final Answer:
The condition so that \( \alpha < k < \beta \) is:
\[ a^2k^2 + abk + ac < 0 \]