Question : If $x=a+\frac{1}{a}$ and $y=a-\frac{1}{a}$, then the value of $x^4+y^4-2x^2y^2$ is:
Option 1: 4
Option 2: 8
Option 3: 16
Option 4: 64
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Correct Answer: 16
Solution :
Given:
$x=a+\frac{1}{a}$, $y=a-\frac{1}{a}$
Now, $x^2=(a+\frac{1}{a})^2=a^2+\frac{1}{a^2}+2$
And $y^2=(a-\frac{1}{a})^2=a^2+\frac{1}{a^2}-2$
Now, we know, $x^4+y^4-2x^2y^2=(x^2-y^2)^2$
$⇒x^4+y^4-2x^2y^2= [(a^2+\frac{1}{a^2}+2)-(a^2+\frac{1}{a^2}-2)]^2$
$\therefore x^4+y^4-2x^2y^2=(2+2)^2=16$
Hence, the correct answer is 16.
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