Question

Consider a quadratic equation $a{z}^2+bz+c=0$, where $a,b,c$ are complex numbers.

$ii)$ If equation has two purely imaginary roots, then which of the following is not true

• A. $a\overline{b}$ is purely imaginary
• B. None of these
• C. $b\overline{c}$ is purely imaginary
• D. $c\overline{a}$ is purely imaginary

Answer 1

The correct option is B None of these

$\left(ii\right)$ Let $z_1,z_2$ be two purely imaginary roots of the

$\Rightarrow z_1=-\overline{z_1},z_2=-\overline{z_2}$
And
$a{z}^2+bz+c=0\dots \left(1\right)$
$\Rightarrow \overline{a{z}^2+bz+c}=0$
$\Rightarrow \overline{a}{\overline{z}}^2+\overline{b}\overline{z}+\overline{c}=0$
$\Rightarrow \overline{a}{z}^2-\overline{b}z+\overline{c}=0\dots \left(2\right)$

Now, equations $\left(1\right)$ and $\left(2\right)$ must be identical as their roots are same
$\Rightarrow \frac{a}{\overline{a}}=\frac{b}{-\overline{b}}=\frac{c}{\overline{c}}$
$\Rightarrow a\overline{c}=\overline{a}c,a\overline{b}+\overline{a}b=0,b\overline{c}+\overline{b}c=0$

Hence, $a\overline{c}$ is purely real and $a\overline{b}$ and $b\overline{c}$ are purely imaginary.