यदि `x=(sqrt(3)+1)/(sqrt(3)-1)` & `y=(sqrt(3)-1)/(sqrt(3)+1)`, है तो `x^(2)+y^(2)` का मान क्या होगा?

Options

A. 14
B. 13
C. 15
D. 10

Answer

A

Explanation

`(sqrt(3)+1)/(sqrt(3)-)` and `y=(sqrt(3)-1)/(sqrt(3)+1)`
`implies :. x=1/y`
`x=(sqrt(3)+1)/(sqrt(3)-1)xx(sqrt(3)+1)/(sqrt(3)+1)`
`=(sqrt(3)+1)^(2)/(3-1)`
`= (3+1+2sqrt(3))/2=(4+2sqrt(3))/2`
`=(2+sqrt(3))`
`x^(2)=(2+sqrt(3))^(2)=4+3+4sqrt(3)`
`=7+4sqrt(3)`
`y^(2)=1/(7+4sqrt(3))xx(7-4sqrt(3))/(7-4sqrt(3))`
`y^(2)=(7-4sqrt(3))/(49-48)=(7-4sqrt(3))/1=`
`7-4sqrt(3)`
`:. x^(2)+y^(2)=7+4sqrt(3)+7-4sqrt(3)`
Alternate:
`x^(2)+y^(2)=x^(2)+1/(x^(2))`
`=(x+1/x)^(2)-2( :. x=1/y)`
`=(sqrt(3)+1)/(sqrt(3)-1)+(sqrt(3)-1)/(sqrt(3)+1^(2))-2`
`=[((sqrt(3)+1)^(2)+(sqrt(3)-1)^(2))/(3-1)]^(2)-2`
`:. (a+b)^(2)+(a-b)^(2)=2(a^(2)+b^(2))`
`=[(2((sqrt(3))^(2)+(1)^(2)))]^(2)-2`
`=(3+1)^(2)-2=16-2=14`