If x and y are positive integer satisfying ${\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}\left(\frac{1}{y}\right)=\frac{1}{7}$, then the number of ordered pairs of $(x,y)$ is

${\tan }^{-1}\left(\frac{1}{x}\right)+{\tan }^{-1}\left(\frac{1}{y}\right)=\frac{1}{7}$

$\Rightarrow {\tan }^{-1}\left(\frac{\frac{1}{x}+\frac{1}{y}}{1-\frac{1}{xy}}\right)={\tan }^{-1}\left(\frac{1}{7}\right)$

$\Rightarrow \frac{x+y}{xy-1}=\frac{1}{7}$

$\Rightarrow 7x+7y=xy-1$

$\Rightarrow (7-y)x=-7y-1$

$\Rightarrow x=\frac{7y+1}{y-7}=7+\frac{50}{y-7}$

Here, $y=$ 8, 9, 12, 17, 32, 57 satisfy above equation.