Question

If roots of the equation $z^2+\alpha z+\beta =0$ lie on $|z|=1$, then

• A. $2|Im\alpha |=1-|\beta {|}^2$
• B. $2|Im\alpha |=|\beta {|}^2-1$
• C. $Im\alpha =0$
• D. None of these

Answer 1

The correct option is D None of these

Let the roots be $z_1$ and $z_2$.

Then $z^2+\alpha z+\beta =(z-z_1)(z-z_2)$

$=z^2-(z_1+z_2)z+z_1{z}_2$

Roots lie on $|z|=1$.

$|z_1{z}_2|=1⟹|\beta |=1$

If $|\beta |=1$, by (A),(B) and (C) $Im\left(\alpha \right)=0$

$Im(-(z_1+z_2\left)\right)=0$

$z_1$ and $z_2$ lie on $|z|=1$

So let $z_1=1,z_2=i$

$\Rightarrow$ Here $Im\left(\alpha \right)=-1\ne 0$

$\therefore$ (A), (B) and (C) are not true.