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x^2 + 4 y^2 = 4 x y

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by sanju09 » Mon Oct 18, 2010 5:57 am
If x^2 + 4 y^2 = 4 x y then (x + y)/ (x - y) is
(A) 2/3
(B) 2
(C) 3
(D) 10/3
(E) 4


[spoiler]???????????[/spoiler]
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by shovan85 » Mon Oct 18, 2010 6:01 am
IMO C

x^2 + 4y^2 - 4xy = 0
=> x^2 - 2 * x * 2y + (2y)^2 = 0
=> (x-2y)^2 = 0
=> x = 2y
=> x/y = 2

Apply Componendo Dividendo:

(x+y)/(x-y) = (2+1)/(2-1) = 3

FYI: https://en.wikipedia.org/wiki/Componendo_and_dividendo
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by fskilnik@GMATH » Mon Oct 18, 2010 6:20 am
shovan85 wrote: x^2 + 4y^2 - 4xy = 0
=> x^2 - 2 * x * 2y + (2y)^2 = 0
=> (x-2y)^2 = 0
=> x = 2y
=> x/y = 2

Apply Componendo Dividendo:

(x+y)/(x-y) = (2+1)/(2-1) = 3

FYI: https://en.wikipedia.org/wiki/Componendo_and_dividendo
Excellent, Shovan.

Please note that substituting x by 2y in the expression asked we find the exact same answer, for sure...
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by shovan85 » Mon Oct 18, 2010 6:26 am
fskilnik wrote:
shovan85 wrote: x^2 + 4y^2 - 4xy = 0
=> x^2 - 2 * x * 2y + (2y)^2 = 0
=> (x-2y)^2 = 0
=> x = 2y
=> x/y = 2

Apply Componendo Dividendo:

(x+y)/(x-y) = (2+1)/(2-1) = 3

FYI: https://en.wikipedia.org/wiki/Componendo_and_dividendo
Excellent, Shovan.

Please note that substituting x by 2y in the expression asked we find the exact same answer, for sure...
Thanks. I did that for cross verification but did not post it here :|
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by fskilnik@GMATH » Mon Oct 18, 2010 6:41 am
shovan85 wrote: I did that for cross verification but did not post it here :|
Sure! I just mentioned it because someone could imagine that this (neat) tool/result you used was essential to get the answer, and of course it was not the case.
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by sandeep800 » Wed Oct 20, 2010 11:27 pm
although at first look i would have tried to solve it by using componendo and dividendo but yeah substitution works well here... !!
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by AtifS » Thu Oct 21, 2010 1:21 am
My explanation:
x^2 + 4y^2 = 4xy
=> x^2 + 4 y^2 - 4xy = 0
=> x^2 - 4 xy + 4 y^2 = 0
=> (x - 2y)^2 = 0
=> x - 2y = 0
=> x = 2y
=> x/2 = y

Now put the value of x/2 into (x + y)/(x - y)

(x + y)/(x - y)
= (x + x/2)/(x - x/2)
= {(2x+x)/2}/{(2x-x)/2}
= (3x/2)/(x/2)
= (3x/2) * (2/x)
= 3x/2 * 2/x ---> 2 will be canceled and x as well (because they are reciprocals of each other)
= 3 --> Answer
So, "C" is the answer for me, which is "3" :)
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by bacchewar_prashant » Thu Oct 21, 2010 1:35 am
C for me
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by frank1 » Thu Oct 21, 2010 7:27 am
sanju09 wrote:If x^2 + 4 y^2 = 4 x y then (x + y)/ (x - y) is
(A) 2/3
(B) 2
(C) 3
(D) 10/3
(E) 4


[spoiler]???????????[/spoiler]
x^2 + 4 y^2 = 4 x y
x^2 + 4 y^2 - 4 x y=0(it is in form a^2+2ab+b^2 where a=x and b=2y)
so
(x-2y)^2=0
x=2y
now actual question is x+y / x -y =?
substitute x=2y

2y+y / 2y-y =3y/y =3

So C
thanks
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by pkit » Fri Oct 22, 2010 2:48 am
Very nice solution, thanks
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