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If x^2+y^2=2xy, then (x+y)/(2x-y)=?

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by Max@Math Revolution » Thu Dec 27, 2018 1:03 am

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[Math Revolution GMAT math practice question]

If x^2+y^2=2xy, then (x+y)/(2x-y)=?

A. -2
B. -1
C. 0
D. 1
E. 2
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by fskilnik@GMATH » Thu Dec 27, 2018 6:10 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x^2+y^2=2xy, then (x+y)/(2x-y)=?

A. -2
B. -1
C. 0
D. 1
E. 2
$$? = {{x + y} \over {2x - y}}$$
$${x^2} + {y^2} = 2xy\,\,\,\left( * \right)$$

Important: x=0 (or y=0) implies (by (*)) that both x and y are equal to zero, what is impossible (because 2x-y is implicitly different from zero).

Conclusion: we may (and will) admit that x and y are nonzero.

$${\left( {x - y} \right)^2} = {x^2} - 2xy + {y^2}\,\,\mathop = \limits^{\left( * \right)} \,\,\,0\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,x - y = 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,x = y$$
$$? = {{2y} \over {2y - y}} = 2\,\,\,\,\,\,\left( {y \ne 0} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{E}} \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Sun Dec 30, 2018 5:17 pm
=>

x^2+y^2=2xy
=> x^2+y^2-2xy=0
=> (x-y)^2 = 0
=> x = y

(x+y)/(2x-y)= (x+x) / (2x-y) = 2x/x = 2, since x =y.


Therefore, the answer is E.
Answer: E
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