The number of positive integral solutions of the equation tan−1⁡x+cos−1⁡y1+y2=sin−1⁡310 is

A
1
B
2
C
0
D
None of these

Solution:

$\begin{array}{l} Given, \tan^{- 1} x + \cos^{- 1} (\frac{y}{\sqrt{1 + y^{2}}}) = \sin^{- 1} (\frac{3}{\sqrt{10}}) \\ \Rightarrow \tan^{- 1} x + \tan^{- 1} (\frac{1}{y}) = \sin^{- 1} (\frac{3}{\sqrt{10}}) \\ \Rightarrow \tan^{- 1} (\frac{x + \frac{1}{y}}{1 - \frac{x}{y}}) = \tan^{- 1} 3 \end{array}$

$\begin{array}{l} \Rightarrow x + \frac{1}{y} = 3 (1 - \frac{x}{y}) \\ \Rightarrow x + \frac{1}{y} = 3 - \frac{3 x}{y} \\ \Rightarrow \frac{1}{y} + \frac{3 x}{y} = 3 - x \\ \Rightarrow \frac{1}{y} (1 + 3 x) = 3 - x \\ ∴ y = \frac{1 + 3 x}{3 - x} \end{array}$

For positive values of r and /, we have
x=1,y=2and x=2,y=7
Hence, the number of solutions of given equation is 2.

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

Solution:

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