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Question : If $x=\frac{1}{x-5}(x>0)$, then the value of $x+\frac{1}{x}$ is:

Option 1: $\sqrt{41}$

Option 2: $\sqrt{29}$

Option 3: $\sqrt{23}$

Option 4: $\sqrt{43}$


Team Careers360 5th Jan, 2024
Answer (1)
Team Careers360 11th Jan, 2024

Correct Answer: $\sqrt{29}$


Solution : $x=\frac{1}{x-5}(x>0)$
$⇒x^2-5x-1=0$
$⇒x^2-1=5x$
Multiplying both sides by $\frac{1}{x}$, we get,
$⇒x-\frac{1}{x}=5$
Squaring both sides, we get,
$⇒(x-\frac{1}{x})^2=5^2$
$⇒x^2+\frac{1}{x^2}-2=25$
Adding 4 to both sides, we get,
$⇒x^2+\frac{1}{x^2}+2=25+4$
$⇒(x+\frac{1}{x})^2=29$
$\therefore x+\frac{1}{x}=\sqrt{29}$
Hence, the correct answer is $\sqrt{29}$.

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