Expansions-13

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If x 3 + y 3 + z 3 = 3 x y z and x + y + z = 0 ; find the value of :

(x + y) 2 / x y + (y + z) 2 / yz + (z + x) 2 / z x

Solution :

x + y + z = 0

=> x + y = - z ,

y + z = - x

and z + x = - y

Therefore, (x + y) 2 / x y + (y + z) 2 / y z + (z + x) 2 / z x = ( - z ) 2 / x y + ( - x ) 2 / y z + ( - y ) 2 / z x

= z 2 / x y + x 2 / y z + y 2 / z x

= ( z 3 + x 3 + y 3 ) / x y z = 3 x y z / x y z = 3

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