if-x-y-z-are-three-distinct-complex-numbers-such-that-x-y-z-y-z-x-z-x-y-0-then-find-the-value-of-x-2-y-z-2-

Question Number 192160 by universe last updated on 10/May/23

if x, y, z are three distinct complex numbers

such that \frac{x}{y - z} + \frac{y}{z - x} + \frac{z}{x - y} = 0 then

find the value of Σ \frac{x^{2}}{{(y - z)}^{2}}

Commented by mehdee42 last updated on 09/May/23

Σ ?

Commented by Frix last updated on 09/May/23

\frac{x}{y - x} + \frac{y}{z - x} + \frac{z}{x - y}, no symmetry, really ?

Commented by Frix last updated on 09/May/23

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

\Rightarrow

\frac{x^{2}}{{(y - z)}^{2}} + \frac{y^{2}}{{(z - x)}^{2}} + \frac{z^{2}}{{(x - y)}^{2}} = 2

Answered by York12 last updated on 09/May/23

\frac{x - z}{y - x} = \frac{y}{x - z} \to {(x - z)}^{2} = y (y - x)

\frac{x - z}{y - x} + \frac{\underline{+} \sqrt{y}}{\underline{+} \sqrt{y - x}} = \frac{\underline{+} 2 \sqrt{y}}{\underline{+} \sqrt{y - x}} = 0 \to y = 0

∴ x = z

∴ \sum_{s y m} \frac{x^{2}}{{(y - z)}^{2}} = 2

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