Question

If $z_1$ and $z_2$ are roots of quadratic equation $a{z}^2+bz+c=0$ such that $Im\left(z_1{z}_2\right)\ne 0$ then

• A. $a,b,c$ are all real
• B. atleast one of $a,b,c$ is real
• C. atleast one of $a,b,c$ is imaginary
• D. $a,b,c$ are all imaginary

Answer 1

Answer: (C)

atleast one of $a,b,c$ is imaginary

$z_1$ and $z_2$ are the roots of quadratic equation $a{z}^2+bz+c=0$.

Then, $z_1{z}_2=\frac{c}{a}$

Since $Im\left(z_1{z}_2\right)\ne 0$, hence $Im\left(\frac{c}{a}\right)\ne 0$.

From this we can say that either $c$ is imaginary or $a$ is imaginary or both are imaginary.