Browse Questions

# If the ratio of the roots of $ax^2+bx+c=0$ is equal to the roots of the equation $x^2+x+1=0$, then $a,b,c$ are in ?

(A) A.P.

(B) G.P.

(C) H.P.

(D) None of these

Toolbox:
• $1+\omega+\omega^2=0$
Let $\alpha,\:\:\beta$ be the roots of $ax^2+bx+c=0$
$\Rightarrow\:\alpha+\beta=\large-\frac{b}{a}$ $, \alpha\beta=\large\frac{c}{a}$
The roots of $x^2+x+1=0$ are $\omega,\:\omega^2$
Given the roots are of equal ratio.
$\Rightarrow\:\large\frac{\alpha}{\beta}=\frac{\omega}{\omega^2}$
$\Rightarrow\:\beta=\alpha\omega$
$\Rightarrow\:\alpha+\beta=\alpha+\alpha\omega=\large-\frac{b}{a}$
$\Rightarrow\:\alpha(1+\omega)=\large-\frac{b}{a}$
$\Rightarrow\:-\alpha\omega^2=-\large\frac{b}{a}$
or $\alpha\omega^2=\large\frac{b}{a}$
or $\alpha=\large\frac{b\omega}{a}$ (since $(\omega^2=\large\frac{1}{\omega})$
and
$\alpha\beta=\alpha^2\omega=\large\frac{c}{a}$
$\Rightarrow\:\large\frac{b^2\omega^2}{a^2}.$$\omega=\large\frac{c}{a}$
$\Rightarrow\:\large\frac{b^2}{a^2}=\frac{c}{a}$ (since $\omega^3=1$)
$or\:\:b^2=ac$
$\therefore\:a,b,c$ are in G.P.