If x+y+z=0 and x^2+y^2+z^2= 12 then find the value of xy+yx+zx?
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Step-by-step explanation:
Use AM-GM inequality.
x^2 + y^2 >(or equal) 2xy
y^2 + z^2 > 2yz
z^2 + x^2 > 2xz
Adding all the equations.
x^2+y^2+z^2-xy-yz-xz >(or equal) 0
Since its given that the equation is equal to zero, we conclude that equation has its minimum value.
So, all the 3 equations should individually attain their minimum values
So,
x^2+y^2-2xy=0
(x-y)^2=0
So, x=y
Similarly, y=z and x=z.
So, x=y=z
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