Question
The number of positive integral solutions of the equation $${\tan ^{ - 1}}x + {\cos ^{ - 1}}\frac{y}{{\sqrt {1 + {y^2}} }} = {\sin ^{ - 1}}\frac{3}{{\sqrt {10} }}$$ is
A.
one
B.
two
C.
zero
D.
None of these
Answer :
two
Solution :
$$\eqalign{
& {\tan ^{ - 1}}x + {\tan ^{ - 1}}\frac{1}{y} = {\tan ^{ - 1}}3\,\,{\text{or,}}\,{\tan ^{ - 1}}\frac{1}{y} = {\tan ^{ - 1}}3 - {\tan ^{ - 1}}x\,\,\,{\text{or, }}{\tan ^{ - 1}}\frac{1}{y} = {\tan ^{ - 1}}\frac{{3 - x}}{{1 + 3x}} \cr
& \Rightarrow \,\,y = \frac{{1 + 3x}}{{3 - x}}. \cr} $$
As $$x, y$$ are positive integers, $$x = 1, 2$$ and corresponding $$y = 2, 7.$$
∴ solutions are $$\left( {x,y} \right) = \left( {1,2} \right),\left( {2,7} \right).$$