If $r=x+y+z$ and ${\tan }^{-1}\sqrt{\frac{xr}{yz}}+{\tan }^{-1}\sqrt{\frac{yr}{zx}}+{\tan }^{-1}\sqrt{\frac{zr}{xy}}=k\pi$then the value of k is

$$\begin{gathered} Let x=y=z=1. \\ \Rightarrow r=3 \\ Then the equation becomes \\ {\tan }^{-1}\sqrt{3}+{\tan }^{-1}\sqrt{3}+{\tan }^{-1}\sqrt{3}=k\mathrm{\pi } \\ \Rightarrow \frac{\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{3}=\mathrm{k\pi } \\ \Rightarrow \mathrm{\pi }=\mathrm{k\pi } \\ \Rightarrow \mathrm{k}=1 \end{gathered}$$