Question

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

• A. $AP$
• B. $GP$
• C. $HP$
• D. $noneofthese$

Answer 1

The correct option is B $GP$

Let $a$ & $b$ be the roots of the equation $\lambda x^2+\mu x+\nu =0$
and $c$ & $d$ be the roots of the equation $x^2+x+1=0$
given that, $\frac{a}{b}=\frac{c}{d}$
By componendo-dividendo $\frac{a+b}{a-b}=\frac{c+d}{c-d}$
$\Rightarrow \frac{{(a+b)}^2}{{(a+b)}^2-4ab}=\frac{{(c+d)}^2}{{(c+d)}^2-4cd}$
$\Rightarrow \frac{{\left(\mu /\lambda \right)}^2}{{\left(\mu /\lambda \right)}^2-4\left(\nu /\lambda \right)}=\frac{1}{1-4}$
$\Rightarrow {\mu }^2=\lambda \nu$
Therefore, $\lambda ,\mu ,\nu$ are in G.P.
Ans: B