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Functions

If x = √3 -...

Question

If $x = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$ , $y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}},$ then the value of $x^{2} + x y + y^{2}$ is

A

5

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B

99

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C

98

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D

none of these

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Solution

The correct option is D $99$
From the given equations we can say, $x = \frac{1}{y}$
$x^{2} + x y + y^{2}$
$= x^{2} + 1 + y^{2}$
$= \frac{1}{y^{2}} + y^{2} + 1$
$= (y + \frac{1}{y})^{2} - 2 + 1$
$= (x + y)^{2} - 1$
$= (3 + 2 - 2 \sqrt{6} + 3 + 2 + 2 \sqrt{6})^{2} - 1$
$= (10)^{2} - 1$
$= 99$

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