Question : If $a=\frac{1}{a - 5}(a>0)$, then the value of $a+\frac{1}{a}$ is:
Option 1: $\sqrt{29}$
Option 2: $–\sqrt{27}$
Option 3: $-\sqrt{29}$
Option 4: $\sqrt{27}$
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Correct Answer: $\sqrt{29}$
Solution :
Given: $a=\frac{1}{a-5}(a>0)$
⇒ $a^{2}-5a=1$
Dividing both sides by $a$ we have,
⇒ $a - 5=\frac{1}{a}$
⇒ $a-\frac{1}{a}=5$
We know that: $(a+\frac{1}{a})^{2}-(a-\frac{1}{a})^{2}=4$
⇒ $(a+\frac{1}{a})^{2}-(5)^{2}=4$
⇒ $(a+\frac{1}{a})^{2}=25+4$
$\therefore(a+\frac{1}{a})=\sqrt{29}$
Hence, the correct answer is $\sqrt{29}$.
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