Question

If $a,b,c$ are in $GP$, where $a,c$ are positive, then the equation $a{x}^2+bx+c=0$ has

• A. real roots
• B. imaginary roots
• C. ratio of roots $1:\omega$ where $\omega$ is nonreal cube root of unity
• D. ratio of roots $b:ac$

Answer 1

Answer: (B), (C)

The correct options are
B imaginary roots
C ratio of roots $1:\omega$ where $\omega$ is nonreal cube root of unity

$a,b,c$ are in G.P and $ac>0$
Therefore, $b^2=ac$ ...(1)
$a{x}^2+bx+c=0$
now, $D=b^2-4ac=ac-4ac=-3ac$
Since, $ac>0$
Therefore, $D<0$ and roots are imaginary.
let $\alpha$ & $\beta$ be the roots of the quadratic equation $a{x}^2+bx+c=0$
$\alpha ,\beta =\frac{-b+\sqrt{b^2-4ac}}{2a}=\frac{-\sqrt{ac}\pm \sqrt{ac-4ac}}{2a}=\sqrt{c}\left(\frac{-1\pm i\sqrt{3}}{2}\right)=\sqrt{c}\omega ,\sqrt{c}{\omega }^2$
Therefore, $\frac{\alpha }{\beta }=1/\omega$
Ans: B,C