Inspired by Siddharth Bhatt

Algebra Level 2

If

x y + y z + z x = 0 , \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 0,

which of the following is true?

Inspiration

x 3 + y 3 + z 3 = 0 x^3 + y^3 + z^3 = 0 x y 2 + y z 2 + z x 2 = 0 xy^2 + yz^2 + zx^2 = 0 x + y + z = 0 x + y + z = 0 x 2 y + y 2 z + z 2 x = 0 x^2 y + y^2z + z^2 x = 0

Chung Kevin by Chung Kevin , 45, Australia

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1 solution

Sravanth C.
Jul 9, 2015

x y + y z + z x = 0 \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 0

Equalizing the denominators, x 2 z y x z + x y 2 z x y + z 2 y x y z = 0 x 2 z + x y 2 + z 2 y x y z = 0 x 2 z + x y 2 + z 2 y = 0 \frac{x^2z}{yxz} + \frac{xy^2}{zxy} + \frac{z^2y}{xyz} = 0 \\ \dfrac{x^2z+xy^2+z^2y}{xyz} = 0 \\ x^2z+xy^2+z^2y=0

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Maybe add a line to explain why x y z ≠ 0 xyz \neq 0 ?

Chung Kevin - 5 years, 11 months ago

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