If `0 lt c lt b lt a` and the roots `alpha, beta` the equation `cx^(2)+bx+a=0` are imaginary, then

Options

A. `(|alpha|+|beta|)/(2)=|alpha||beta|`
B. `(1)/(|alpha|)=(1)/(|beta|)`
C. `(1)/(|alpha|)+(1)/(|beta|)lt2`
D. `(1)/(|alpha|)+(1)/(|beta|)gt 2`

Answer

To solve the problem, we need to analyze the conditions given in the question and use the properties of quadratic equations and their roots.

### Step-by-Step Solution:

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

2. **Rearranging the Inequality**:
Rearranging the inequality gives us:
\[ b^2 < 4ac \]

3. **Using the Given Conditions**:
We know from the problem statement that \( 0 < c < b < a \). Since \( c \), \( b \), and \( a \) are all positive, we can use this information to analyze the inequality \( b^2 < 4ac \).

4. **Analyzing the Inequality**:
Since \( c < b < a \), we can substitute \( a \) with a value greater than \( b \) and \( c \) with a value less than \( b \). This implies:
\[ 4ac > 4bc \quad \text{(since \( a > b \) and \( c < b \))} \]
Therefore, we need to check if \( b^2 < 4bc \).

5. **Dividing by \( b \) (since \( b > 0 \))**:
Dividing both sides of \( b^2 < 4bc \) by \( b \) gives:
\[ b < 4c \]

6. **Conclusion**:
We have established that \( b < 4c \) under the conditions provided. Now, we need to analyze the options given in the question to find the correct one.

### Analyzing the Options:
1. **Option 1**: \( \frac{|\alpha| + |\beta|}{2} = |\alpha| + |\beta| \) - This is not possible as it implies both roots are zero.
2. **Option 2**: \( \frac{1}{|\alpha|} = \frac{1}{|\beta|} \) - This implies \( |\alpha| = |\beta| \), which is possible since both roots are conjugates.
3. **Option 3**: \( \frac{1}{|\alpha|} + \frac{1}{|\beta|} < 2 \) - This could be true if both roots are greater than 1.
4. **Option 4**: \( \frac{1}{|\alpha|} + \frac{1}{|\beta|} > 2 \) - This is not possible as it contradicts the condition of the roots being imaginary.

### Final Answer:
The correct option is **Option 3**: \( \frac{1}{|\alpha|} + \frac{1}{|\beta|} < 2 \).