If `0 gt a gt b gt c` and the roots `alpha, beta` are imaginary roots of `ax^(2) + bx + c = 0` then

Options

A. `|alpha|= |beta|`
B. `|alpha| gt 1`
C. `|beta| lt 1`
D. `alpha+beta=0`

Answer

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

### Step-by-Step Solution:

1. **Understand the Condition for Imaginary Roots**:
For the roots of the quadratic equation \( ax^2 + bx + c = 0 \) to be imaginary, the discriminant must be less than zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Therefore, we need:
\[ b^2 - 4ac < 0 \]

2. **Analyze the Given Inequalities**:
We are given that \( 0 > a > b > c \). This means that \( a, b, c \) are all negative numbers.

3. **Substituting the Values**:
Since \( a, b, c < 0 \), we can analyze the product \( ac \):
- Since both \( a \) and \( c \) are negative, \( ac > 0 \) (the product of two negative numbers is positive).

4. **Evaluating the Discriminant**:
Since \( b < 0 \) and \( a, c < 0 \), we can say:
\[ b^2 > 0 \quad \text{and} \quad 4ac > 0 \]
Therefore, \( b^2 - 4ac < 0 \) holds true, confirming that the roots are indeed imaginary.

5. **Expressing the Roots**:
The roots can be expressed in the form:
\[ \alpha = A + Bi \quad \text{and} \quad \beta = A - Bi \]
where \( A \) is the real part and \( B \) is the imaginary part.

6. **Finding the Magnitudes**:
The magnitude of \( \alpha \) is given by:
\[ |\alpha| = \sqrt{A^2 + B^2} \]
Similarly, the magnitude of \( \beta \) is:
\[ |\beta| = \sqrt{A^2 + B^2} \]
Since both magnitudes are equal, we have:
\[ |\alpha| = |\beta| \]

7. **Conclusion**:
From the analysis, we find that the magnitudes of the imaginary roots \( \alpha \) and \( \beta \) are equal. Therefore, the correct option is:
\[ \text{Option A: } |\alpha| = |\beta| \]