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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