​ If x = √3 - √2/√3 + √2 and y = √3 + √2/√3 - √2, then x^2 + xy + y^2 = A. 101 B. 99 C. 98 D. 102 ​

Question

If x = \(\frac{\sqrt3-\sqrt2}{\sqrt3+\sqrt2}\) and y = \(\frac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\), then x² + xy + y² =

A. 101

B. 99

C. 98

D. 102

Answer 1

Given x = \(\frac{\sqrt3-\sqrt2}{\sqrt3+\sqrt2}\), y = \(\frac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\) 

x² = (5 – 2√6)² = 25 +24 -20√6) = 49 – 20√6

Similarily y = \(\frac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\) x \(\frac{\sqrt3+\sqrt2}{\sqrt3+\sqrt2}\) = 5 + 2√6

y² = (5 + 2√6 )² = 49 + 20√6

xy = (5 - 2√6) ( 5 + 2√6) = 25 – 24 = 1

So, x²+ xy+y² = 49 – 20√6 +1 + 49 + 20√6 = 99