If $$x = a + \frac{1}{a}$$   and $$y = a - \frac{1}{a},$$   then the value of x⁴ + y⁴ - 2x²y² is?

A. 4

B. 8

C. 16

D. 64

Answer: Option C

Solution(By Examveda Team)

$$\eqalign{ & x = a + \frac{1}{a} \cr & {x^2} = {a^2} + \frac{1}{{{a^2}}} + 2 \cr & {y^2} = {a^2} + \frac{1}{{{a^2}}} - 2 \cr & {\text{Now, }} \cr & {x^4} + {y^4} - 2{x^2}{y^2} \cr & = {\left( {{x^2} - {y^2}} \right)^2} \cr & = {\left( {{a^2} + \frac{1}{{{a^2}}} + 2 - {a^2} - \frac{1}{{{a^2}}} + 2} \right)^2} \cr & = {\left( 4 \right)^2} \cr & = 16 \cr} $$