Question

Let $\alpha$ and $\beta$ be the real numbers and $z$ be a complex number. If $x^2+\alpha z+\beta =0$ has two distinct non-real roots with $Re\left(x\right)=1$, 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

Answer: (C)

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

Let roots be $p+iq$ and $p-iq,p,qϵR$

root lie on line $Re\left(z\right)=1$

$\Rightarrow p=1$

Product of roots $=p^2+q^2=\beta =1+q^2$

$\Rightarrow \beta ϵ(1,\infty ),(q\ne 0,\because$ roots are distinct).