Question : If $(x+\frac{1}{x}):(x-\frac{1}{x})=5 : 3$, then the value(s) of $x$ is/are:
Option 1: $\pm 1$
Option 2: $\pm 2$
Option 3: $\pm 3$
Option 4: $0$
Correct Answer: $\pm 2$
Solution :
Given: $(x+\frac{1}{x}):(x-\frac{1}{x})=5:3$
By using Componendo and Dividendo rule, $a:b = c:d ⇒ \frac{a+b}{a–b}=\frac{c+d}{c–d}$,
where $a=x,b=\frac{1}{x},c=5,$ and $d=3$
By putting the given value of a, b, c, d in the formula, we get:
$\frac{x+\frac{1}{x}+x-\frac{1}{x}}{x+\frac{1}{x}-x+\frac{1}{x}}=\frac{5+3}{5-3}$
⇒ $\frac{2x}{\frac{2}{x}}=\frac{8}{2}$
⇒ $x^{2}=4$
$\therefore x=\pm 2$
Hence, the correct answer is $\pm2$.
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