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 $\frac{4}{5}$
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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