How should I attack it?

Algebra Level 3

If x > y > 0 x>y>0 and it satisfies the equation x + y x − y = 2 \frac{x+y}{x-y} = \sqrt 2 , then what is the value of x 2 + y 2 x y \frac{x^2+y^2}{xy} ?

NMTC 2012


The answer is 6.

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Harshi Singh by Harshi Singh , 20, India

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3 solutions

Square both sides:

x 2 + 2 x y + y 2 x 2 − 2 x y + y 2 = 2 \dfrac{x^2+2xy+y^2}{x^2-2xy+y^2}=2

Now, clear denominators and try to obtain the wanted expression:

x 2 + 2 x y + y 2 = 2 x 2 − 4 x y + 2 y 2 x^2+2xy+y^2=2x^2-4xy+2y^2

6 x y = x 2 + y 2 6xy=x^2+y^2

x 2 + y 2 x y = 6 \dfrac{x^2+y^2}{xy}=\boxed{6}

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Rwit Panda
Jun 11, 2015

Squaring both the sides, we get 2=(x^2 +y^2 + 2xy)/(x^2 +y^2 - 2xy). Now, using componendo-dividendo, we get 2(x^2 + y^2)/4xy = 3. So we obtain (x^2 + y^2)/xy = 6.

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Madhukar Thalore
Jun 11, 2015

better solution apply componendo and dividendo on both sides to get a relation b /w x and y.Then just put the value of either in the expression you will get the answer the relation is x=(3+2root2)y

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I dont know the concept of componendo and dividendo can you explain it a little bit @Madhukar Thalore

Harshi Singh - 6 years ago

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yes i can explain actually what you need to do is as follows:- say if

a+b/a-b=1/6

then add numerator and denominator on both sides simultaneously and put the final expression in numerator and similarly subtract denominator from numerator and put it in denominator ( you can do vice versa too ) to get

2a/2b=7/-5

which can be simplified to get

a/b=-7/5

hence u got a relation easily. Thanks :) Harshi Singh

Madhukar Thalore - 6 years ago

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