Question : If $a+\frac{1}{a-2}=4$, then the value of $(a-2)^{2}+(\frac{1}{a-2})^{2}$ is:
Option 1: 0
Option 2: 2
Option 3: –2
Option 4: 4
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Correct Answer: 2
Solution :
Given: $a+\frac{1}{a-2}=4$
Subtracting 2 from both sides,
⇒ $a-2+\frac{1}{a-2}=2$
Squaring both sides,
⇒ $( a-2)^2+\frac{1}{( a-2)^2} + 2=2^2$
⇒ $( a-2)^2+\frac{1}{( a-2)^2}=2^2-2$
⇒ $( a-2)^2+\frac{1}{( a-2)^2}=2$
Hence, the correct answer is 2.
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