Let `alpha,beta` be real and z be a complex number. If `z^2+alphaz""+beta=""0` has two distinct roots on the line Re `z""=""1` , then it is necessary that : (1) `b"" in (0,""1)` (2) `b"" in (-1,""0)` (3) `|b|""=""1` (4) `b"" in (1,oo)`

Explanation

To solve the problem, we need to analyze the given quadratic equation \( z^2 + \alpha z + \beta = 0 \) under the condition that it has two distinct roots on the line where the real part of \( z \) is equal to 1.

### Step-by-Step Solution:

1. **Identify the Roots**:
Since the roots are on the line \( \text{Re}(z) = 1 \), we can express the roots as:
\[
z_1 = 1 + i y \quad \text{and} \quad z_2 = 1 - i y
...