Question : If $3\sqrt{\frac{1-a}{a}}+9=19-3\sqrt{\frac{a}{1-a}};$ then, what is the value of $a?$
Option 1: $\frac{3}{10}$ and $\frac{7}{10}$
Option 2: $\frac{1}{10}$ and $\frac{9}{10}$
Option 3: $\frac{2}{5}$ and $\frac{3}{5}$
Option 4: $\frac{1}{5}$ and $\frac{4}{5}$
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Correct Answer: $\frac{1}{10}$ and $\frac{9}{10}$
Solution : Put $\sqrt{\frac{(1 - a)}{a}} = x$ ⇒ $3x + 9 = 19 - \frac{3}{x}$ ⇒ $3x + \frac{3}{x} = 10$ ⇒ $x = 3, \frac{1}{3}$ Now, $\sqrt{\frac{(1 - a)}{a}} = 3$ ⇒ $\frac{(1 - a)}{a}= 9$ ⇒ $10a = 1$ ⇒ $a = \frac{1}{10}$ And $\sqrt{\frac{(1 - a)}{a}} =\frac{1}{3}$ ⇒ $\frac{(1 - a)}{a}= \frac{1}{9}$ ⇒ $10a = 9$ ⇒ $a = \frac{9}{10}$ $\therefore a =\frac{1}{10}$ and $\frac{9}{10}$ Hence, the correct answer is $\frac{1}{10}$ and $\frac{9}{10}$.
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