The number of solutions of \({\text{tan}}^{−1}4x+{\text{tan}}^{−1}6x=\frac{\pi }{6}\), where \(−\frac{1}{2\sqrt{6}} [JEE…

\({\text{tan}}^{−1}4x+{\text{tan}}^{−1}6x=\frac{\pi }{6}\)

\(\Rightarrow {\text{tan}}^{−1}\left(\frac{4x+6x}{1-24{x}^{2}}\right)=\frac{\pi }{6}\)

\(\Rightarrow \frac{10x}{1−24{x}^{2}}=\frac{1}{\sqrt{3}}\)

\(\Rightarrow 24{x}^{2}+10\sqrt{3}x−1=0\)

\(x=\frac{−10\sqrt{3}\pm \sqrt{300+96}}{48}\)

\(x=\frac{\sqrt{396}−10\sqrt{3}}{48}\)

Only 1 solution in \(\left(−\frac{1}{2\sqrt{6}},\frac{1}{2\sqrt{6}}\right)\)