If the roots of the equation `ax^2 + bx + c = 0, a != 0 (a, b, c` are real numbers), are imaginary and `a + c < b,` then

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) under the conditions that the roots are imaginary and \( a + c < b \).

### Step-by-step Solution:

1. **Understanding the Condition for Imaginary Roots**:
The roots of a quadratic equation are imaginary if the discriminant \( D \) is less than zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
For the roots to be imaginary:
\[ b^2 - 4ac < 0 \]

2. **Analyzing the Condition \( a + c < b \)**:
We are given that \( a + c < b \). We can rearrange this inequality:
\[ b > a + c \]

3. **Combining the Conditions**:
We need to check if both conditions can coexist. From the first condition, we have:
\[ b^2 < 4ac \]
To explore the implications of \( a + c < b \), we can substitute \( b \) with \( a + c + k \) where \( k > 0 \):
\[ (a + c + k)^2 < 4ac \]
Expanding this gives:
\[ a^2 + 2ac + c^2 + 2k(a + c) + k^2 < 4ac \]
Rearranging this leads to:
\[ a^2 + c^2 + k^2 + 2k(a + c) < 2ac \]

4. **Conclusion**:
The inequality \( a^2 + c^2 + k^2 + 2k(a + c) < 2ac \) suggests that for \( k > 0 \), the left side can be less than the right side under certain conditions. Thus, the conditions \( b^2 < 4ac \) and \( a + c < b \) can coexist, confirming that the roots of the equation are indeed imaginary under the given constraints.

### Final Statement:
Therefore, if the roots of the equation \( ax^2 + bx + c = 0 \) are imaginary and \( a + c < b \), then the conditions are consistent with the properties of quadratic equations.