Question

If $\alpha$ is a complex constant such that $\alpha z^2+z+\overline{\alpha }=0$ has a real root, then

• A. $\alpha +\overline{\alpha }=1$
• B. $\alpha +\overline{\alpha }=0$
• C. $\alpha +\overline{\alpha }=-1$
• D. The absolute value of the real root is 1

Answer 1

Answer: (A), (C), (D)

The correct options are
A $\alpha +\overline{\alpha }=1$
C $\alpha +\overline{\alpha }=-1$
D The absolute value of the real root is 1

Let $z=c$ be a real root. Then,
$a{c}^2+c+\overline{a}=0$ (1)
Putting $\alpha =p+iq$, we have
$(p+iq)c^2+c+p-iq=0$
$\Rightarrow p{c}^2+c+p=0$ and $q{c}^2-q=0\Rightarrow c=\pm 1(\because q\ne 0)$
$\therefore \left(1\right)\Rightarrow \alpha \pm 1+\overline{\alpha }=0$
Also, $|c|=1$