Question Number 171826 by Mikenice last updated on 21/Jun/22
$${if}\:{a}+{b}+{c}=\mathrm{0},{find}\:{the}\:{value}\:{of}\: \\ $$$$\frac{{a}^{\mathrm{2}} }{{bc}}+\frac{{b}^{\mathrm{2}} }{{ca}}+\frac{{c}^{\mathrm{2}} }{{ab}} \\ $$
Answered by nimnim last updated on 21/Jun/22
$$\frac{{a}^{\mathrm{2}} }{{bc}}+\frac{{b}^{\mathrm{2}} }{{ca}}+\frac{{c}^{\mathrm{2}} }{{ab}} \\ $$$$=\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }{{abc}} \\ $$$$=\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }{{abc}}−\mathrm{3}+\mathrm{3} \\ $$$$=\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} −\mathrm{3}{abc}}{{abc}}+\mathrm{3} \\ $$$$=\frac{\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right)}{{abc}}+\mathrm{3} \\ $$$$=\frac{\left(\mathrm{0}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right)}{{abc}}+\mathrm{3} \\ $$$$=\mathrm{0}+\mathrm{3} \\ $$$$=\mathrm{3} \\ $$$$ \\ $$
Commented by Mikenice last updated on 21/Jun/22
$${thanks}\:{sir} \\ $$