Question

The number of positive integral solutions of the equation ${\tan }^{-1}x+{\cot }^{-1}y={\tan }^{-1}3,$ is

• A. two
• B. one
• C. infinite
• D. None of these

Answer 1

Answer: (A)

${\tan }^{-1}x+{\tan }^{-1}\frac{1}{y}={\tan }^{-1}3$
$\Rightarrow {\tan }^{-1}\frac{x+\frac{1}{y}}{1-\frac{x}{y}}={\tan }^{-1}3\Rightarrow \frac{xy+1}{y-x}=3$
$\Rightarrow y=\frac{1+3x}{3-x}>0$ [$\because$ x and y are positive]
$\Rightarrow x-3<0\Rightarrow x<3$ or $x=1,2$
$\therefore y=2,7$ solution set is
$\left(x,y\right)\in \left\{\left(1,2\right),\left(2,7\right)\right\}$