Let `p` and `q` are complex numbers such that `|p|+|q| lt 1`. If `z_(1)` and `z_(2)` are the roots of the `z^(2)+pz+q=0`, then which one of the following is correct ?

Options

A. `|z_(1)| lt 1` and `|z_(2)| lt 1`
B. `|z_(1)| gt 1` and `|z_(2)| gt 1`
C. If `|z_(1)| lt 1`, then `|z_(2)| gt 1` and vice versa
D. Nothing definite can be said

Explanation

To solve the problem, we need to analyze the quadratic equation given by \( z^2 + pz + q = 0 \) and the properties of its roots \( z_1 \) and \( z_2 \).

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots of the quadratic equation \( z^2 + pz + q = 0 \) can be expressed using Vieta's formulas:
- Sum of the roots: \( z_1 + z_2 = -p \)
- Product of the roots: \( z_1 z_2 = q \)

2. **Take Modulus of the Roots**:
Taking the modulus of the sum of the roots:
\[ |z_1 + z_2| = |-p| = |p| \]
Taking the modulus of the product of the roots:
\[ |z_1 z_2| = |q| \]

3. **Use the Given Condition**:
We know from the problem that:
\[ |p| + |q| < 1 \]

4. **Apply the Triangle Inequality**:
From the triangle inequality, we have:
\[ |z_1 + z_2| \leq |z_1| + |z_2| \]
Substituting the value from step 2:
\[ |p| \leq |z_1| + |z_2| \]

5. **Express \( |z_1| + |z_2| \)**:
We can also express \( |z_1| + |z_2| \) in terms of \( |z_1 z_2| \):
\[ |z_1 + z_2| = |p| \quad \text{and} \quad |z_1 z_2| = |q| \]
Thus, we can write:
\[ |z_1| + |z_2| \geq |p| - |z_1 z_2| = |p| - |q| \]

6. **Combine the Inequalities**:
From the earlier step, we have:
\[ |p| \leq |z_1| + |z_2| \quad \text{and} \quad |z_1| + |z_2| \geq |p| - |q| \]
Combining these gives:
\[ |p| + |q| < 1 \]

7. **Conclude the Modulus of Roots**:
Since \( |p| + |q| < 1 \), we can conclude that:
\[ |z_1| < 1 \quad \text{and} \quad |z_2| < 1 \]

### Final Result:
Thus, we can conclude that both roots \( z_1 \) and \( z_2 \) have moduli less than 1.