Question Number 191395 by Mingma last updated on 23/Apr/23
Answered by Rasheed.Sindhi last updated on 23/Apr/23
$${xy}+{yz}+{zx}=\mathrm{0};\:\:\frac{{y}+{z}}{{x}}+\frac{{x}+{z}}{{y}}+\frac{{x}+{y}}{{z}}=? \\ $$$$\frac{{y}+{z}}{{x}}+\frac{{x}+{z}}{{y}}+\frac{{x}+{y}}{{z}} \\ $$$$=\frac{{y}+{z}}{{x}}+\mathrm{1}+\frac{{x}+{z}}{{y}}+\mathrm{1}+\frac{{x}+{y}}{{z}}+\mathrm{1}−\mathrm{3} \\ $$$$=\frac{{x}+{y}+{z}}{{x}}+\frac{{x}+{y}+{z}}{{y}}+\frac{{x}+{y}+{z}}{{z}}−\mathrm{3} \\ $$$$=\left({x}+{y}+{z}\right)\left(\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\right)−\mathrm{3} \\ $$$$=\left({x}+{y}+{z}\right)\left(\frac{{xy}+{yz}+{zx}}{{xyz}}\right)−\mathrm{3} \\ $$$$=\left({x}+{y}+{z}\right)\left(\frac{\mathrm{0}}{{xyz}}\right)−\mathrm{3}=−\mathrm{3} \\ $$