If $\frac{\tan \mathrm{x}}{2}=\frac{\tan \mathrm{y}}{3}=\frac{\tan \mathrm{z}}{5},\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{\pi }$ and ${\tan }^2\mathrm{x}+{\tan }^2\mathrm{y}+{\tan }^2\mathrm{z}=\frac{38}{\mathrm{K}}$ then K is

 tan x = 2t, tan y = 3t, tan z = 5t 

$\begin{array}{l}\text{ Also }\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{\pi }\\ \therefore \tan \mathrm{x}+\tan \mathrm{y}+\tan \mathrm{z}=\tan \mathrm{xtan}\mathrm{ytan}\mathrm{z}\\ \Rightarrow {\mathrm{t}}^2=\frac{1}{3}\\ \Rightarrow {\tan }^2\mathrm{x}+{\tan }^2\mathrm{y}+{\tan }^2\mathrm{z}={\mathrm{t}}^2(4+9+25)=38{\mathrm{t}}^2,\\ \therefore \mathrm{K}=3\end{array}$