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\} $