Let λ ≠ 0 be in R . If α and β are the roots of the equation x 2 - x + 2 λ = 0 ,…

$\alpha +\beta =1 ...\left(\mathrm{i}\right)$

$\alpha \beta =2\lambda ...\left(\mathrm{ii}\right)$
$\alpha +\gamma =\frac{10}{3} ...\left(\mathrm{iii}\right)$
$\alpha \gamma =9\lambda ...\left(\mathrm{iv}\right)$
from $\left(\mathrm{iv}\right) \& \left(\mathrm{ii}\right)$
$\frac{\gamma }{\beta }=\frac{9}{2} ...\left(\mathrm{v}\right)$

from $\left(\mathrm{iii}\right)-\left(\mathrm{i}\right)$

$\gamma -\beta =\frac{10}{3}-1=\frac{7}{3} ...\left(\mathrm{vi}\right)$

from $\left(\mathrm{v}\right)$ and $\left(\mathrm{vi}\right)$

$\beta =\frac{2}{3},\gamma =3$
from $\left(\mathrm{i}\right) \alpha =1-\beta =1-\frac{2}{3}=\frac{1}{3}$
from $\left(\mathrm{ii}\right) \lambda =\frac{\alpha \beta }{2}=\frac{\frac{1}{3}\times \frac{2}{3}}{2}=\frac{1}{9}$
So $\frac{\beta \gamma }{\lambda }=\frac{\frac{2}{3}\times 3}{\frac{1}{9}}=18$