Question
If $$z_1, z_2$$ are the roots of the quadratic equation $$az^2 + bz + c = 0$$ such that $$\operatorname{Im} \left( {{z_1},{z_2}} \right) \ne 0$$ then
A.
$$a, b, c$$ are all real
B.
at least one of $$a, b, c$$ is real
C.
at least one of $$a, b, c$$ is imaginary
D.
all of $$a, b, c$$ are imaginary
Answer :
at least one of $$a, b, c$$ is imaginary
Solution :
$${\text{Since}}\,\,a{z^2} + bz + c = 0\,\,\,.....\left( 1 \right)$$
and $$z_1 , z_2$$ (roots of (1)) are such that $$\operatorname{Im} \left( {{z_1},{z_2}} \right) \ne 0.$$
Now, $$z_1$$ and $$z_2$$ are not conjugates of each other
Complex roots of (1) are not conjugate of each other
Co-efficient $$a, b, c$$ can-not all be real at least one $$a, b, c$$ is imaginary.