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

$\alpha z^2+z+\overline{\alpha }=0Letxbearealrooti.e.x=\overline{x}\alpha x^2+x+\overline{\alpha }=0...\left(1\right)Conjugatingeq.\left(1\right)weget,\overline{\alpha }{\overline{x}}^2+\overline{x}+\alpha =0⟹\overline{\alpha }{x}^2+x+\alpha =0...\left(2\right)Subtractingeq.\left(1\right)\&eq.\left(2\right)weget,x^2=1⟹x=\pm 1\therefore \left|x\right|=1Multiplyingeq.\left(1\right)by\overline{\alpha }andeq.\left(2\right)by\alpha andthensubtractingweget,x=-(\alpha +\overline{\alpha })=\pm 1$
Ans: A,C &D