Question
If X = (√5 + 1) / (√5 – 1) and
y = (√5 – 1) / (√5 + 1) then the value of (x 2+xy+y 2) / (x 2 – xy + y 2) isSolution
X = (√5 + 1) / (√5 – 1) and y = (√5 – 1) / (√5 + 1) then (x2+xy+y2) / (x2 – xy + y2) 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 = x2 + y2 + 2xy (3)2 = x2 + y2 + 2 => x2 + y2 = 7 => Put x2 + y2 = 7 and xy = 1 in the question => (7 +1) / (7 – 1) => 8/6 = 4/3
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