If $x = a^{1 / 2} + a^{- 1 / 2}, y = a^{1 / 2} - a^{- 1 / 2}$ , then value of $(x^{4} - x^{2} y^{2} - 1) + (y^{4} - x^{2} y^{2} + 1)$ .

9

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25

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14

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16

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None of these

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Solution

The correct option is D $16$
$x = a^{1 / 2} + a^{- 1 / 2} . . . . . (1) y = a^{1 / 2} - a^{- 1 / 2} . . . . . (2) t h e n (x^{4} - x^{2} y^{2} - 1) + (y^{4} - x^{2} y^{2} + 1) x^{4} - x^{2} y^{2} + 1 + y^{4} - x^{2} y^{2} + 1 x^{4} + y^{4} - 2 x^{2} y^{2} {(x^{2})}^{2} + {(y^{2})}^{2} - 2 x^{2} y^{2} {(x^{2} - y^{2})}^{2} [\begin{matrix} ∵ x^{2} - y^{2} = (x - y) (x + y) x^{2} - 2 x y + y^{2} = {(x - y)}^{2} \end{matrix}] {[(x - y) (x + y)]}^{2} = ? . . . . . . . . (3) A d d i n g a n d s u b s t r a c t i n g b o t h e q u a t i o n r e s p e c t i v e l y A d d i n g \frac{\begin{matrix} x = a^{\frac{1}{2}} + a^{\frac{- 1}{2}} y = a^{\frac{1}{2}} - a^{\frac{- 1}{2}} \end{matrix}}{x + y = 2 a^{\frac{1}{2}}} S u b s t r a c t i n g \frac{\begin{matrix} x = a^{\frac{1}{2}} + a^{\frac{- 1}{2}} y = a^{\frac{1}{2}} - a^{\frac{- 1}{2}} \end{matrix}}{x - y = 2 a^{- \frac{1}{2}}} p u t t i n g t h e v a l u e o f (x + y) a n d (x - y) i n e q (3) {[(2 a^{\frac{- 1}{2}}) (2 a^{\frac{1}{2}})]}^{2} {[4 a^{0}]}^{2} = 16 [∵ a^{0} = 1]$

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