If `x=(sqrt(3)-\ sqrt(2))/(sqrt(3)+sqrt(2))` and `y=(sqrt(3)+\ sqrt(2))/(sqrt(3)-\ sqrt(2))` , then `x^2+\ x y+y^2=` 101 (b) 99 (c) 98 (d) 102

Explanation

Solution:
`x= ((sqrt(3)- sqrt(2))/(sqrt(3)+sqrt(2)))`
Rationalizing
`x= [(sqrt(3)- sqrt(2))/(sqrt(3)+sqrt(2))] xx [(sqrt(3)- sqrt(2)) / (sqrt(3) - sqrt(2))]`
`=>(sqrt(3) - sqrt(2))^(2)/(2)`
`y =((sqrt(3)+ sqrt(2))/(sqrt(3)-sqrt(2)) =(sqrt(3) + sqrt(2))^(2)/ 1)`
`x^2+\ x y+y^2=(sqrt(3)- sqrt(2))^(4) + (sqrt(3)+ sqrt(2))^(4)+(sqrt(3)+ sqrt(2))^(2) xx (sqrt(3)+ sqrt(2))^(2)`
`(3 +2 -2sqrt(6))^(2)+(3 +2 +2sqrt(6))^(2)+(3-2)^(2)`
Property used
`(a-b)^2=a^(2)- b^(2)`
`(5 -2sqrt(6))^(2) = (5 -2sqrt(6))^(2) + 1`
`25-4sqrt(6) xx 5 + 24 +25 5 xx 4sqrt(6) +24`
`= 49 + 49`
`=98`
The correct answer is option `(c)`