looking for some shortcut or faster method for solving a question involving square root

Question

If $x =\dfrac{ \sqrt 3 - \sqrt 2 }{ \sqrt 3 + \sqrt 2 } $ and $ y = \dfrac{\sqrt 3 + \sqrt 2 }{ \sqrt 3 - \sqrt 2}$, find the value of $\dfrac{x^2 + xy + y^2}{x^2 - xy + y^2}.$

I obviously know the traditional method of solving this qn (by multiplying by conjugate). I want to know if there is some faster method to solve this question.

Answer 1

$$\implies xy=1$$

and $$x=\dfrac{(\sqrt3-\sqrt2)^2}{3-2}=5-2\sqrt6$$

Similarly, $y=5+2\sqrt6$

$$x^2+xy+y^2=(x+y)^2-xy=?$$

Answer 2

Use $$x^2+xy+y^2=(x+y)^2-xy$$ and $$x^2-xy+y^2=(x-y)^2+xy$$ and you can easily find $x+y,x-y,xy$

Complete solution: $$x=5-2√6$$ $$y=5+2√6$$ $$xy=1$$ Answer=$$\frac{10^2-1}{(-4√6)^2+1}$$$$=\frac{99}{97}$$

Answer 3

We get $x+y=10,xy=1$

$\dfrac{x^2 + xy + y^2}{x^2 - xy + y^2}=\dfrac{(x+y)^2-xy}{(x+y)^2-3xy}=\dfrac{99}{97}$