If-ax-by-cz-0-and-a-2-x-b-2-y-c-2-z-0-Find-the-ratio-x-y-z-

Question Number 7612 by Tawakalitu. last updated on 06/Sep/16

$${If}\:\:{ax}\:+\:{by}\:+\:{cz}\:=\:\mathrm{0},\: \\ $$$${and},\:\:{a}^{\mathrm{2}} {x}\:+\:{b}^{\mathrm{2}} {y}\:+\:{c}^{\mathrm{2}} {z}\:=\:\mathrm{0} \\ $$$${Find}\:{the}\:{ratio}\:\:{x}:{y}:{z} \\ $$

Commented by Rasheed Soomro last updated on 06/Sep/16

ax + by + cz = 0.................(i) a^2 x + b^2 y + c^2 z = 0.............(ii) x=((−by−cz)/a) x=((−b^2 y−c^2 z)/a^2 ) ((−by−cz)/a)=((−b^2 y−c^2 z)/a^2 ) −aby−acz=−b^2 y−c^2 z b^2 y−aby=−c^2 z+acz b(b−a)y=c(a−c)z (y/z)=((c(a−c))/(b(b−a))) y : z = c(a−c) : b(b−a) ..........(iii) Again from (i) & (ii) z=((−ax−by)/c) z=(( −a^2 x −b^2 y)/c^2 ) ((−ax−by)/c)=(( −a^2 x −b^2 y)/c^2 ) −acx−bcy=−a^2 x −b^2 y a^2 x−acx=bcy−b^2 y a(a−c)x=b(c−b)y (x/y)=((b(c−b))/(a(a−c))) x : y = b(c−b) : a(a−c)...................(iv) By combining (iv) & (iii) ((x,:,y,:,z),((b(c−b)),:,(a(a−c)),,),(,,(c(a−c)),:,(b(b−a))),((bc(c−b)),:,(ca(a−c)),:,(ab(b−a))) )

$${ax}\:+\:{by}\:+\:{cz}\:=\:\mathrm{0}……………..\left({i}\right) \\ $$$$\:{a}^{\mathrm{2}} {x}\:+\:{b}^{\mathrm{2}} {y}\:+\:{c}^{\mathrm{2}} {z}\:=\:\mathrm{0}………….\left({ii}\right) \\ $$$${x}=\frac{−{by}−{cz}}{{a}} \\ $$$${x}=\frac{−{b}^{\mathrm{2}} {y}−{c}^{\mathrm{2}} {z}}{{a}^{\mathrm{2}} } \\ $$$$\frac{−{by}−{cz}}{{a}}=\frac{−{b}^{\mathrm{2}} {y}−{c}^{\mathrm{2}} {z}}{{a}^{\mathrm{2}} } \\ $$$$−{aby}−{acz}=−{b}^{\mathrm{2}} {y}−{c}^{\mathrm{2}} {z} \\ $$$${b}^{\mathrm{2}} {y}−{aby}=−{c}^{\mathrm{2}} {z}+{acz} \\ $$$${b}\left({b}−{a}\right){y}={c}\left({a}−{c}\right){z} \\ $$$$\frac{{y}}{{z}}=\frac{{c}\left({a}−{c}\right)}{{b}\left({b}−{a}\right)} \\ $$$${y}\::\:{z}\:=\:{c}\left({a}−{c}\right)\::\:{b}\left({b}−{a}\right)\:\:……….\left({iii}\right) \\ $$$${Again}\:{from}\:\left({i}\right)\:\&\:\left({ii}\right) \\ $$$$\:{z}=\frac{−{ax}−{by}}{{c}} \\ $$$${z}=\frac{\:−{a}^{\mathrm{2}} {x}\:−{b}^{\mathrm{2}} {y}}{{c}^{\mathrm{2}} } \\ $$$$\frac{−{ax}−{by}}{{c}}=\frac{\:−{a}^{\mathrm{2}} {x}\:−{b}^{\mathrm{2}} {y}}{{c}^{\mathrm{2}} } \\ $$$$−{acx}−{bcy}=−{a}^{\mathrm{2}} {x}\:−{b}^{\mathrm{2}} {y} \\ $$$${a}^{\mathrm{2}} {x}−{acx}={bcy}−{b}^{\mathrm{2}} {y} \\ $$$${a}\left({a}−{c}\right){x}={b}\left({c}−{b}\right){y} \\ $$$$\frac{{x}}{{y}}=\frac{{b}\left({c}−{b}\right)}{{a}\left({a}−{c}\right)} \\ $$$${x}\::\:{y}\:=\:{b}\left({c}−{b}\right)\::\:{a}\left({a}−{c}\right)……………….\left({iv}\right) \\ $$$${By}\:{combining}\:\left({iv}\right)\:\&\:\left({iii}\right) \\ $$$$\begin{pmatrix}{{x}}&{:}&{{y}}&{:}&{{z}}\\{{b}\left({c}−{b}\right)}&{:}&{{a}\left({a}−{c}\right)}&{}&{}\\{}&{}&{{c}\left({a}−{c}\right)}&{:}&{{b}\left({b}−{a}\right)}\\{{bc}\left({c}−{b}\right)}&{:}&{{ca}\left({a}−{c}\right)}&{:}&{{ab}\left({b}−{a}\right)}\end{pmatrix} \\ $$

Commented by Tawakalitu. last updated on 06/Sep/16