If `k in R` lies between the roots of `ax^(2) + 2b x + c =0` , then

Options

A. `ak^(2) + 2bk + c lt 0`
B. `a^(2) k^(2) + 2abk + ac lt 0`
C. `a^(2) k^(2) + 2abk + ac gt 0`
D. `ak^(2) + 2bk + c gt 0`

Answer

To solve the problem where \( k \in \mathbb{R} \) lies between the roots of the quadratic equation \( ax^2 + 2bx + c = 0 \), we will analyze the conditions based on the sign of \( a \).

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots of the quadratic equation \( ax^2 + 2bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \( A = a \), \( B = 2b \), and \( C = c \). Thus, the roots \( \alpha \) and \( \beta \) are:
\[ \alpha, \beta = \frac{-2b \pm \sqrt{(2b)^2 - 4ac}}{2a} = \frac{-2b \pm \sqrt{4b^2 - 4ac}}{2a} = \frac{-b \pm \sqrt{b^2 - ac}}{a} \]

2. **Condition for \( k \) to be Between the Roots**:
For \( k \) to lie between the roots \( \alpha \) and \( \beta \), we need:
\[ \alpha < k < \beta \quad \text{or} \quad \beta < k < \alpha \]
This implies that the quadratic function \( f(k) = ak^2 + 2bk + c \) must be negative at \( k \):
\[ f(k) < 0 \]

3. **Case 1: \( a > 0 \)**:
- If \( a > 0 \), the parabola opens upwards. The function \( f(x) \) is negative between the roots.
- Since \( k \) lies between the roots, we have:
\[ f(k) = ak^2 + 2bk + c < 0 \]

4. **Case 2: \( a < 0 \)**:
- If \( a < 0 \), the parabola opens downwards. The function \( f(x) \) is positive between the roots.
- Since \( k \) lies between the roots, we have:
\[ f(k) = ak^2 + 2bk + c > 0 \]

5. **Conclusion**:
- If \( k \) lies between the roots of the quadratic equation, then:
- If \( a > 0 \), \( ak^2 + 2bk + c < 0 \)
- If \( a < 0 \), \( ak^2 + 2bk + c > 0 \)

### Final Result:
Thus, the condition for \( k \) lying between the roots of the quadratic equation \( ax^2 + 2bx + c = 0 \) depends on the sign of \( a \):
- If \( a > 0 \): \( ak^2 + 2bk + c < 0 \)
- If \( a < 0 \): \( ak^2 + 2bk + c > 0 \)