Question

Find the number of possible positive integral solution which satisfy the equation ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1-y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

${\tan }^{-1}x+{\tan }^{-1}\frac{1}{y}={\tan }^{-1}3\Rightarrow {\tan }^{-1}\frac{1}{y}={\tan }^{-1}3-{\tan }^{-1}x$

$\Rightarrow {\tan }^{-1}\frac{1}{y}={\tan }^{-1}\left(\frac{3-x}{1+3x}\right)\Rightarrow y=\left(\frac{1+3x}{3-x}\right)$

As $x,y$ are positive integers, $x=1,2$ and corresponding $y=2,7$

Therefore solutions are $(x,y)=(1,2),(2,7)$