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Level 1

I f x + y + z = 0 t h e n w h a t i s t h e v a l u e o f 1 x 2 + y 2 − z 2 + 1 y 2 + z 2 − x 2 + 1 z 2 + x 2 − y 2 ? If\quad x+y+z=0\quad then\quad what\quad is\quad the\quad value\quad of\\ \frac { 1 }{ { x }^{ 2 }+{ y }^{ 2 }-{ z }^{ 2 } } +\frac { 1 }{ { y }^{ 2 }+{ z }^{ 2 }-{ x }^{ 2 } } +\frac { 1 }{ { z }^{ 2 }+{ x }^{ 2 }-{ y }^{ 2 } } ?


The answer is 0.

Satyajit Ghosh by Satyajit Ghosh , 21, India

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1 solution

Ronak Agarwal
Aug 22, 2014

S = 1 x 2 + y 2 − z 2 + 1 y 2 + z 2 − x 2 + 1 z 2 + x 2 − y 2 S=\frac { 1 }{ { x }^{ 2 }+{ y }^{ 2 }-{ z }^{ 2 } } +\frac { 1 }{ { y }^{ 2 }+{ z }^{ 2 }-{ x }^{ 2 } } +\frac { 1 }{ { z }^{ 2 }+{ x }^{ 2 }-{ y }^{ 2 } }

In first term use z = − x − y z=-x-y , in second term use x = − y − z x=-y-z , in third term use y = − x − z y=-x-z to get :

S = 1 − 2 x y + 1 − 2 y z + 1 − 2 z x S=\frac { 1 }{ -2xy } +\frac { 1 }{ -2yz } +\frac { 1 }{ -2zx }

⇒ S = − 1 2 ( x + y + z x y z ) \Rightarrow S=\frac { -1 }{ 2 } (\frac { x+y+z }{ xyz } )

Now since x + y + z = 0 x+y+z=0 we have :

S = 0 \boxed{S=0}

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