Question

If the ratio of the roots of $\lambda x^2+\mu x+v=0$ is equal to the ratio of the roots of $x^2+x+1=0$, then $\lambda ,\mu ,v$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (B)

Let $\alpha ,\beta$ be the roots of $\lambda x^2+\mu x+v=0$
$\therefore \alpha +\beta =-\frac{\mu }{\lambda },a\beta =\frac{v}{\lambda }$
$\therefore \frac{(\alpha +\beta {)}^2}{\alpha \beta }=\frac{\frac{{\mu }^2}{{\lambda }^2}}{\frac{v}{\lambda }}=\frac{{\mu }^2}{\lambda v}$
$\Rightarrow \frac{\alpha }{\beta }+\frac{\beta }{\alpha }+2=\frac{{\mu }^2}{\lambda v}(1)$
Let $\gamma ,\delta$ be the roots of $x^2+x+1=0$
$\therefore \gamma +\delta =-1,\gamma \delta =1$
$\therefore \frac{(\gamma +\delta {)}^2}{\gamma \delta }=1$
$\Rightarrow \frac{\gamma }{\delta }+\frac{\delta }{\gamma }+2=1(2)$
Since $\frac{\alpha }{\beta }=\frac{\gamma }{\delta }$
$\therefore$ from Eq. (1) and (2)
$\frac{{\mu }^2}{\lambda v}=1$
$\Rightarrow {\mu }^2=\lambda v$
$\therefore \lambda ,\mu ,v$ are in G.P.