If x = a + 1 a , y = a - 1 a , ( a > 0 ) then the value of x4 + y4 - 2x2y2 is?

Answer: Option A

$$\begin{array}{rl}& x=\sqrt{a}+\frac{1}{\sqrt{a}}\\ & y=\sqrt{a}-\frac{1}{\sqrt{a}}\\ & \text{Put }a=4\\ & x=2+\frac{1}{2}=\frac{5}{2}\\ & y=2-\frac{1}{2}=\frac{3}{2}\\ & \text{Then, }{x}^4+y^4-2x^2{y}^2\\ & ={\left(x^2-y^2\right)}^2\\ & ={\left(\frac{25}{4}-\frac{9}{4}\right)}^2\\ & =\text{ 16}\end{array}$$