Let p, q, r `in` R and `r gt p gt 0`. If the quadratic equation `px^(2) + qx + r = 0` has two complex roots `alpha and beta`, then `|alpha|+|beta|`, is

Options

A. less than 2 but not equal to 1
B. equal to 2
C. equal to 1
D. greater than 2

Explanation

To solve the problem, we need to analyze the quadratic equation given by \( px^2 + qx + r = 0 \) under the conditions that \( r > p > 0 \) and that the roots \( \alpha \) and \( \beta \) are complex.

### Step-by-Step Solution:

1. **Understanding the Nature of Roots**:
Since the roots are complex, the discriminant of the quadratic equation must be negative. The discriminant \( D \) is given by:
\[ D = q^2 - 4pr \]
For the roots to be complex, we need:
\[ D < 0 \implies q^2 < 4pr \]

2. **Using the Properties of Roots**:
The roots \( \alpha \) and \( \beta \) can be expressed in terms of their product and sum:
- The sum of the roots \( \alpha + \beta = -\frac{q}{p} \)
- The product of the roots \( \alpha \beta = \frac{r}{p} \)

3. **Finding the Modulus of the Roots**:
Since \( \alpha \) and \( \beta \) are complex, we can express them in terms of their modulus:
\[ \alpha = a + i b \quad \text{and} \quad \beta = a - i b \]
where \( a \) and \( b \) are real numbers. The modulus of the roots is given by:
\[ |\alpha| = \sqrt{a^2 + b^2} \quad \text{and} \quad |\beta| = \sqrt{a^2 + b^2} \]

4. **Relating Modulus to Product**:
The product of the roots can be expressed as:
\[ |\alpha| \cdot |\beta| = |\alpha|^2 = \frac{r}{p} \]
Since \( r > p > 0 \), we have:
\[ |\alpha|^2 = \frac{r}{p} > 1 \implies |\alpha| > 1 \]
Similarly, since \( \beta \) is the conjugate of \( \alpha \), we have:
\[ |\beta|^2 = \frac{r}{p} > 1 \implies |\beta| > 1 \]

5. **Summing the Moduli**:
Now, we can add the moduli:
\[ |\alpha| + |\beta| > 1 + 1 = 2 \]

### Conclusion:
Thus, we conclude that:
\[ |\alpha| + |\beta| > 2 \]