IfX = (√5 + 1) / (√5 – 1) and y = (√5 – 1) / (√5 + 1) then the value of(x ^{2}+xy+y ^{2}) / (x ^{2} – xy + y ^{2}) is

X = (√5 + 1) / (√5 – 1) and y = (√5 – 1) / (√5 + 1) then (x^{2}+xy+y^{2}) / (x^{2} – xy + y^{2}) firstly we will find x+y X + y = (√5 + 1) / (√5 – 1) + (√5 – 1) / (√5 + 1) => ((√5 + 1)^{ 2} + (√5 – 1)^{2}) / (5 – 1) => ( 5+1 +2√5+5+1-2√5) / 4 => (12 / 4) = 3 x × y = (√5 + 1) / (√5 – 1) × (√5 – 1) / (√5 + 1) = 1 => ((x + y)^{2} = x^{2} + y^{2} + 2xy (3)^{2} = x^{2} + y^{2} + 2 => x^{2} + y^{2 }= 7 => Put x^{2} + y^{2 }= 7 and xy = 1 in the question => (7 +1) / (7 – 1) => 8/6 = 4/3

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