Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 173369 by AgniMath last updated on 10/Jul/22

Answered by mr W last updated on 10/Jul/22

((by+cz)/(b^2 +c^2 ))=((cz+ax)/(c^2 +a^2 ))=((ax+by)/(a^2 +b^2 ))=k, say  ax+by=k(a^2 +b^2 )    ...(i)  by+cz=k(b^2 +c^2 )     ...(ii)  cz+ax=k(c^2 +a^2 )   ...(iii)  Σ:  ax+by+cz=k(a^2 +b^2 +c^2 )   ...(iv)  (iv)−(ii):  ⇒ax=ka^2  ⇒(x/a)=k  similarly  ⇒(y/b)=k, (z/c)=k  ⇒(a/x)=(b/b)=(c/z)=k

$$\frac{{by}+{cz}}{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }=\frac{{cz}+{ax}}{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} }=\frac{{ax}+{by}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }={k},\:{say} \\ $$$${ax}+{by}={k}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)\:\:\:\:...\left({i}\right) \\ $$$${by}+{cz}={k}\left({b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)\:\:\:\:\:...\left({ii}\right) \\ $$$${cz}+{ax}={k}\left({c}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)\:\:\:...\left({iii}\right) \\ $$$$\Sigma: \\ $$$${ax}+{by}+{cz}={k}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)\:\:\:...\left({iv}\right) \\ $$$$\left({iv}\right)−\left({ii}\right): \\ $$$$\Rightarrow{ax}={ka}^{\mathrm{2}} \:\Rightarrow\frac{{x}}{{a}}={k} \\ $$$${similarly} \\ $$$$\Rightarrow\frac{{y}}{{b}}={k},\:\frac{{z}}{{c}}={k} \\ $$$$\Rightarrow\frac{{a}}{{x}}=\frac{{b}}{{b}}=\frac{{c}}{{z}}={k} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com