If $$3\sqrt {\frac{{1 - a}}{a}} + 9 = 19 - 3\sqrt {\frac{a}{{1 - a}}} ,$$      then what is the value of a?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Put }}\sqrt {\frac{{1 - a}}{a}} = x \cr & \Rightarrow 3x + 9 = 19 - \frac{3}{x} \cr & \Rightarrow 3x + \frac{3}{x} = 10 \cr & \Rightarrow x = 3,\,\frac{1}{3} \cr & {\text{Now, }}\sqrt {\frac{{1 - a}}{a}} = 3 \cr & \Rightarrow \frac{{1 - a}}{a} = 9 \cr & \Rightarrow 10a = 1 \cr & \Rightarrow a = \frac{1}{{10}} \cr & {\text{and }}\sqrt {\frac{{1 - a}}{a}} = \frac{1}{3} \cr & \Rightarrow \frac{{1 - a}}{a} = \frac{1}{9} \cr & \Rightarrow 10a = 9 \cr & \Rightarrow a = \frac{9}{{10}} \cr & \therefore \,a = \frac{1}{{10}}\,{\text{and }}\frac{9}{{10}} \cr} $$