Let $z_{1}, z_{2}, z_{3}$ be three distinct complex numbers and $α, β, γ$ are three positive real numbers such that
$\frac{α}{| z_{2} - z_{3} |} = \frac{β}{| z_{3} - z_{1} |} = \frac{γ}{| z_{1} - z_{2} |} t h e n \frac{α^{2}}{(z_{2} - z_{3})} + \frac{β^{2}}{(z_{3} - z_{1})} + \frac{γ^{2}}{(z_{1} - z_{2})} =_____$

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$\bar{z_{1}} + \bar{z_{2}} + \bar{z_{3}}$

$z_{1} z_{2} z_{3}$

$z_{1} + z_{2} + z_{3}$

answer is C.

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Detailed Solution

$\frac{α^{2}}{{| z_{2} - z_{3} |}^{2}} = \frac{β^{2}}{{| z_{3} - z_{2} |}^{2}} = \frac{γ^{2}}{{| z_{1} - z_{2} |}^{2}} = λ$

$∴ \frac{α^{2}}{(z_{2} - z_{3})} + \frac{β^{2}}{(z_{3} - z_{1})} + \frac{γ}{(z_{1} - z_{3})}$

$= λ (\bar{z_{2}} - \bar{z_{3}} + \bar{z_{3}} - \bar{z_{1}} + \bar{z_{1}} - \bar{z_{2}})$

$= λ (0) = 0$

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