Question : If $a^2+1=9a$ and $a\neq0$, then the value of $a^2+\frac{1}{a^2}$ is:
Option 1: 81
Option 2: 18
Option 3: 79
Option 4: 83
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Correct Answer: 79
Solution :
Given:
$a^2+1=9a, a\neq0$
Dividing both sides by $a$, we get
$a+\frac{1}{a}=9$
Squaring both sides, we get
$a^2+\frac{1}{a^2}+2×a×\frac{1}{a}=81$
⇒ $a^2+\frac{1}{a^2}=81-2$
$\therefore a^2+\frac{1}{a^2}=79$
Hence, the correct answer is 79.
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