Question

Let $\alpha$ and $\beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Re\left(z\right)=1$, then it is necessary that

• A. $\beta \in (-1,0)$
• B. $\left|\beta \right|=1$
• C. $\beta \in [1,\infty )$
• D. $\beta \in (0,1)$

Answer 1

The correct option is C

$\beta \in [1,\infty )$

Explanation for correct answer

Assume a complex number $z=x+\$iy$ given that $Re\left(z\right)=1$
The roots are complex and also conjugate with one another.
$z=1+iy$ and $1-iy$ are the two roots of $z^2+\alpha z+\beta =0$
Product of the roots $\beta$

$$\begin{gathered} (1+iy)(1-iy)=\beta \\ \beta =1+y^2\ge 1 \\ \beta \in [1,\infty ) \end{gathered}$$

Hence, the correct answer is (c)