Find the value
If \[ x = \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} + \sqrt{2}}, \quad y = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} – \sqrt{2}}, \] then \[ x^2 + xy + y^2 = \ ? \]
Solution:
\[ x = \frac{(\sqrt{3} – \sqrt{2})^2}{3 – 2} = 5 – 2\sqrt{6} \]
\[ y = \frac{(\sqrt{3} + \sqrt{2})^2}{3 – 2} = 5 + 2\sqrt{6} \]
\[ x + y = 10, \quad xy = 1 \]
\[ x^2 + y^2 = (x + y)^2 – 2xy = 100 – 2 = 98 \]
\[ x^2 + xy + y^2 = 98 + 1 = 99 \]
\[ \therefore \text{Correct Answer: } 99 \ (\text{Option b}) \]