If $\left(\mathrm{\alpha },\mathrm{\alpha }-1\right)$ lies inside the ellipse $16{\mathrm{x}}^2+9{\mathrm{y}}^2–16\mathrm{x} = 0$then $\mathrm{\alpha }$ lies in the interval $\left(\frac{9}{25},\mathrm{K}\right),$ then the value of K =

$$\begin{gathered} {\mathrm{S}}_{11}<0 \\ 16{\mathrm{\alpha }}^2+9{(\mathrm{\alpha }-1)}^2-16\mathrm{\alpha }<0 \\ 25{\mathrm{\alpha }}^2-18\mathrm{\alpha }+9-16\mathrm{\alpha }<0 \\ 25{\mathrm{\alpha }}^2-34\mathrm{\alpha }+9<0 \\ 25{\mathrm{\alpha }}^2-25\mathrm{\alpha }-9\mathrm{\alpha }+9<0 \\ (25\mathrm{\alpha }-9) (\mathrm{\alpha }-1)<0 \end{gathered}$$

$\underset{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$25$}\right. 1}{\overset{+ - +}{\_\_\_\_\_\_\_\_}}$

$\mathrm{\alpha }\in \left(\frac{9}{25}, 1\right)$

By comparision k=1