यदि `x=(sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3))` और `y=(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))` है तो `(x^(2)+xy+y^(2))/(x^(2)-xy+y^(2))=?`

Options

A. `65/63`
B. `67/65`
C. `69/67`
D. `63/61`

Answer

D

Explanation

Given
`x=(sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3)),y=(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))`
Find: `(x^(2)+y^(2)+xy)/(x^(2)+y^(2)-xy)=?`
`implies (x^(2)+y^(2)+2xy-xy)/(x^(2)+y^(2)-2xy+xy)`
`((x+y)^(2)-xy)/((x-y)^(2)+xy)=?`
Now `x+y=((sqrt(5)-sqrt(3)))/((sqrt(5)+sqrt(3)))+((sqrt(5)-sqrt(3)))/((sqrt(5)-sqrt(3)))`
`implies x+y=(2(sqrt(5)^(2)+sqrt(3)^(2)))/2`
`implies x+y=8`...........i
Again `x=y=((sqrt(5)-sqrt(3))(sqrt(5)+sqrt(3)))/((sqrt(5)+sqrt(3))(sqrt(5)-sqrt(3)))`
`=(4xxsqrt(5)xxsqrt(3))/2`
`implies (x-y)=2sqrt(15)`.............ii
And `xy=((sqrt(5)-sqrt(3))/((sqrt(5)+sqrt(3)))xx(sqrt(5)+sqrt(3)))/((sqrt(5)-sqrt(3)))`
`xy=1`
Substitutes values in the question.
`implies ((x+y)^(2)-xy)/((x=y)^(2)+xy)`
`=(8^(2)-1)/((2sqrt(15))^(2)+1)implies 63/61`