Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta=0$ has two distinct roots on…
Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re} z=1$, then it is necessary that
$\beta \in(-1,0)$
$|\beta|=1$
$\beta \in(1, \infty)$
$\beta \in(0,1)$
Solution
Suppose roots are $1+\mathrm{pi}, 1+\mathrm{qi}$
Sum of roots $1+p i+1+q i=-\alpha$ which is real $\Rightarrow$ roots of $1+\mathrm{pi}, 1-\mathrm{pi}$
Product of roots $=\beta=1+p^2 \in(1, \infty)$
$\mathrm{p} \neq 0$ since roots are distinct.