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प्रश्न
If x = `((sqrt(3) + 1))/((sqrt(3) - 1)` and y = `((sqrt(3) - 1))/((sqrt(3) - 1)`, find the values of
x2 - y2 + xy
उत्तर
x2 - y2 + xy
x2 - y2 + xy = (x + y) (x - y) + xy ----(1)
∴ (x + y) = `((sqrt(3) + 1))/((sqrt(3) - 1)) + ((sqrt(3) - 1))/((sqrt(3) + 1)`
= `((sqrt(3) + 1)^2 + (sqrt(3) - 1)^2)/(3 - 1)`
= `(3 + 1 + 2sqrt(3) + 3 + 1 - 2sqrt(3))/(2)`
= `(8)/(2)`
= 4
(x - y) = `((sqrt(3) + 1))/((sqrt(3) - 1)) xx ((sqrt(3) - 1))/((sqrt(3) + 1)`
= `((sqrt(3) + 1)^2 - (sqrt(3) - 1)^2)/(3 - 1)`
= `(3 + 1 + 2sqrt(3) - 3 - 1 + 2sqrt(3))/(2)`
= `2sqrt(3)`
and xy = `((sqrt(3) + 1))/((sqrt(3) - 1)) xx ((sqrt(3) - 1))/((sqrt(3) + 1)`
= `(3 - 1)/(3 - 1)`
= 1
substitutingin (1), we get
x2 - y2 + xy
= (x+ y) (x - y) + xy
= `4 xx 2sqrt(3) + 1`
= `8sqrt(3) + 1`
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