Question : If $x=a^{\frac{1}{2}}+a^{-\frac{1}{2}}$, $y=a^{\frac{1}{2}}- a^{-\frac{1}{2}}$, then the value of $(x^{4}-x^{2}y^{2}-1)+(y^{4}-x^{2}y^{2}+1)$ is:
Option 1: 16
Option 2: 13
Option 3: 12
Option 4: 14
Correct Answer: 16
Solution :
Given: $x=a^{\frac{1}{2}}+a^{\frac{-1}{2}}$ and $y=a^{\frac{1}{2}}-a^{\frac{-1}{2}}$
⇒ $x=\sqrt{a}+{\frac{1}{\sqrt{a}}}$ and $y=\sqrt{a}-{\frac{1}{\sqrt{a}}}$
⇒ $x+y=2\sqrt{a}$ and $x-y=\frac{2}{\sqrt{a}}$
So, $x^{4}-x^{2}y^{2}-1+y^{4}-x^{2}y^{2}+1$
= $x^{4}+y^{4}-2x^{2}y^{2}$
= $(x^{2}-y^{2})^{2}$
= $[(x-y)(x+y)]^{2}$
= $[(2\sqrt{a})(\frac{2}{\sqrt{a}})]^{2}$
= $16$
Hence, the correct answer is 16.
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