Question Number 211367 by mr W last updated on 07/Sep/24
$${solve}\:{for}\:{R}^{+} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −{kxy}={c}^{\mathrm{2}} \\ $$$${y}^{\mathrm{2}} +{z}^{\mathrm{2}} −{kyz}={a}^{\mathrm{2}} \\ $$$${z}^{\mathrm{2}} +{x}^{\mathrm{2}} −{kzx}={b}^{\mathrm{2}} \\ $$$$\left({k}\:{is}\:{constant}\right) \\ $$
Commented by ajfour last updated on 11/Sep/24
https://youtu.be/LgXNcnsOJtM?si=JMlLjooG3OzelvXT
Answered by a.lgnaoui last updated on 09/Sep/24
![k+2=(((x+y)^2 −c^2 )/(xy))=(((y+z)^2 −a^2 )/(yz))=(((x+z)^2 −b^2 )/(xz)) alors: xy=(((x+y)^2 )/(k+2)) −(c^2 /(k+2))(1) yz=(((y+z)^2 )/(k+2))−(a^2 /(k+2)) (2) xz=(((x+z)^2 )/(k+2))−(b^2 /(k+2))(3) 2(x^2 +y^2 +z^2 )−k(xy+yz+xz)=a^2 +b^2 +c^2 4(x^2 +y^2 +z^3 )−k[(x+y+z)^2 −(x^2 +y^2 +z^2 )]=2(a+b+c) (4+k)(x^2 +y^2 +z^2 )−k(x+y+z)^2 =2(a^2 +b^2 +c^2 ) or x^2 +y^2 +z^2 =(x+y+z)^2 −2(xy+yz+xz) ⇒ 3(x+y+z)^2 =2(a^2 +b^2 +c^2 ) x+y+z=(√((2/3)(a^2 +b^2 +c^2 ) )) (i) x+y=(√((2/3)(a^2 +b^2 +c^2 ))) −z (a^2 −b^2 )=(y^2 −x^2 )−kz(y−x) =(y−x)[(x+y)−kz] soit a^2 −b^2 =(y−x)[(√((2/3)(a^2 +b^2 +c^2 ))) −z−kz] y−x=((a^2 −b^2 )/( [(√((2/3)(a^2 +b^2 +c^3 ))) −(k+1)z])) { ((y=((a^2 −b^2 )/(2[(√((2/3)e))−(k+1)z])) −(z/2)+(1/2)(√((2/3)e)))),((e=(√(a^2 +b^2 +c^2 )))) :} a^2 =(y−(k/2)z)^2 +(((4−k^2 )/4))z^2 y=(k/2)z+(√(a^2 −(((4−k^2 )/4))z^2 )) donc on a lequation en z z^2 (√(a^2 −((4−k^2 )/4))) + z(((k+1)/2))−((a^2 −b^2 )/(2[(√((2/3)e)) −(k+1)z])) =(1/2)(√((2/3)e)) this equation is long othere methode: −−−−−−−−− ..(i)⇒ (√((2/3)e)) −z=(x+y) k+2=(((x+y)^2 −c^2 )/(xy))=(((y+z)^2 −a^2 )/(yz)) ⇒ ((x+y)−c)/x)=((y+z)−a)/z) ⇒a^2 + ((z[(x+y)−c^2 ])/x)=(y+z) ((x+y)−c)/x)=((y+z)−a)/z)=(((x+y)^2 −(y+z)^2 +a^2 −c^2 )/(x−z)) (((x−z)(x+z+2y)+)/(x−z))=(((x+z+2y))/1)+(((a−c)/(x−z))) (y+z)^2 =((z[(x+y)^2 −c^2 ])/x)+a^2 y+z=((k/2)+1)z+(√(a^2 −((4−k^2 )/4)z^2 )) (((y+z)^2 −a^2 )/(yz))=k+2 ⇒(y+z)^2 =(k+2)yz+a^2 =a^2 +(k+2)z((k/2)z+(√(a^2 −((4−k^2 )/4)z^2 )) ) donc (z+(k/2)z+(√(a^3 −((4−k^2 )/4)z^2 )) )^2 = a^2 +(((k+2)k)/2)z^2 +(k+2)z(√(a−((4−k^2 )/4)z^2 )) . soit apres calculs ((((k+2)^2 )/4)z^2 −((k(k+2))/2)z^2 )−((4−k)/4)z^2 =0 (((k+2)^2 −2(k+2))/4)=((4−k)/4) k^2 +3k−4=0 △=25 { ((k=+1)),((k=−4)) :} on choisit k>0 k=1 3=(((x+y)^2 −c^2 )/(xy)) =(((y+z)^2 −a^2 )/(yz)) (x+y)^2 −3xy−c^2 =0 (x−(3/2)y)^2 −((9/4)y^2 +c^2 )=0 (ii) (y−(3/2)z)^2 −((9/4)z^2 +c^2 )=0 (iii) y=(2/2)z+(√(a−(3/4)z^2 )) ⇒(a−(3/4)z^2 )=c^2 +(9/4)z^2 3z^2 =a^2 −c^2 z=(√((a^2 −c^2 )/3)) { (((x+y)=(√((2/3)(a^2 +b^2 +c^2 ))) −(√((a^2 −c^2 )/3)) )),((xy =(((x+y)^2 −c^2 )/3)=(((2/3)(a+b+c)+((a^2 −c^2 )/3)−(2/3)(√(2a^2 −c^2 )(a^2 +b^2 +c^2 )) −c^2 )/3))) :} { ((xy=((2/9)(√(a+b+c)) )[((√(a^2 +b^2 +c^2 )) −(√(a^2 −c^2 )) )]+((a^2 −4c^2 )/9))),((x+y =(√((2/3)(a^2 +b^2 +c^2 )) −(√((a^2 −c^2 )/3)))) :} { ((k^2 −sk+p=0)),((s=(x+y) p=xy)) :} k=((s±(√(s^2 −4p)))/2) s^(2 ) =3xy+c^2 p=xy ⇒ k=((3xy±(√(9xy−4xy)))/2) =((3xy+(√(5xy)))/2)=(1/2)(√(xy)) [3(√(xy)) +5] { (((√x)=(1/2)((√y) (3(√(xy)) +5) )),(((√y) =(1/2)(√x) (3(√(xy)) −5))) :} (√(xy)) =(1/4)(√(xy)) (3(√(xy)) +5)(3(√(xy)) −5) 1=(1/4)(3m+5)(3m−5) m=(√(xy)) 9m^2 −25=4 m=((√(29))/3) (√(xy)) =((√(29))/3) (x+y=(√(c^2 +3xy)) ⇒ x+y=(√(c^2 +((29)/3))) { ((x+y =(√(c^2 +((29)/3))))),((xy =((29)/9))) :} { ((△=((3c^2 +29)/3)−((116)/9))),( { ((x=(((1/3)((√(27c^2 −29)) −(√(9c^2 +29)) ))/6))),((y=(((1/3)((√(27c^2 −29)) +(√(9c^2 +29)) ))/6))) :}) :} Recap k=1 x=(((√(27c^2 −29)) −(√(9c^2 +29)))/(18)) y=(((√(27c^2 −29)) +(√(9c^2 +29)))/(18)) z=((√(a^2 −c^2 ))/3) pour ( k=−4): (meme procedure)](https://www.tinkutara.com/question/Q211389.png)
$$\mathrm{k}+\mathrm{2}=\frac{\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{2}} −\mathrm{c}^{\mathrm{2}} }{\mathrm{xy}}=\frac{\left(\mathrm{y}+\mathrm{z}\right)^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} }{\mathrm{yz}}=\frac{\left(\mathrm{x}+\mathrm{z}\right)^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} }{\mathrm{xz}} \\ $$$$\mathrm{alors}: \\ $$$$\mathrm{xy}=\frac{\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{2}} }{\mathrm{k}+\mathrm{2}}\:−\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{k}+\mathrm{2}}\left(\mathrm{1}\right) \\ $$$$\mathrm{yz}=\frac{\left(\mathrm{y}+\mathrm{z}\right)^{\mathrm{2}} }{\mathrm{k}+\mathrm{2}}−\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{k}+\mathrm{2}}\:\left(\mathrm{2}\right)\:\:\:\:\:\mathrm{xz}=\frac{\left(\mathrm{x}+\mathrm{z}\right)^{\mathrm{2}} }{\mathrm{k}+\mathrm{2}}−\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{k}+\mathrm{2}}\left(\mathrm{3}\right) \\ $$$$ \\ $$$$\:\:\:\mathrm{2}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{z}}^{\mathrm{2}} \right)−\boldsymbol{\mathrm{k}}\left(\boldsymbol{\mathrm{xy}}+\boldsymbol{\mathrm{yz}}+\boldsymbol{\mathrm{xz}}\right)=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \\ $$$$\:\:\:\mathrm{4}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{z}}^{\mathrm{3}} \right)−\boldsymbol{\mathrm{k}}\left[\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} −\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{z}}\:^{\mathrm{2}} \right)\right]=\mathrm{2}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}\right) \\ $$$$ \\ $$$$ \\ $$$$\left(\mathrm{4}+\boldsymbol{\mathrm{k}}\right)\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{z}}^{\mathrm{2}} \right)−\boldsymbol{\mathrm{k}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} =\mathrm{2}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right) \\ $$$$ \\ $$$$\boldsymbol{\mathrm{or}}\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{z}}^{\mathrm{2}} =\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} −\mathrm{2}\left(\boldsymbol{\mathrm{xy}}+\boldsymbol{\mathrm{yz}}+\boldsymbol{\mathrm{xz}}\right) \\ $$$$\Rightarrow \\ $$$$\:\:\mathrm{3}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} =\mathrm{2}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right) \\ $$$$ \\ $$$$\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}=\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)\:}\:\left(\mathrm{i}\right) \\ $$$$ \\ $$$$ \\ $$$$\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)}\:−\boldsymbol{\mathrm{z}}\:\:\: \\ $$$$ \\ $$$$\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} \right)=\left(\boldsymbol{\mathrm{y}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)−\boldsymbol{\mathrm{kz}}\left(\boldsymbol{\mathrm{y}}−\boldsymbol{\mathrm{x}}\right) \\ $$$$\:\:=\left(\boldsymbol{\mathrm{y}}−\boldsymbol{\mathrm{x}}\right)\left[\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)−\boldsymbol{\mathrm{kz}}\right] \\ $$$$\boldsymbol{\mathrm{soit}} \\ $$$$\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} =\left(\boldsymbol{\mathrm{y}}−\boldsymbol{\mathrm{x}}\right)\left[\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)}\:−\boldsymbol{\mathrm{z}}−\boldsymbol{\mathrm{kz}}\right] \\ $$$$ \\ $$$$ \\ $$$$\boldsymbol{\mathrm{y}}−\boldsymbol{\mathrm{x}}=\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} }{\:\left[\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{3}} \right)}\:−\left(\boldsymbol{\mathrm{k}}+\mathrm{1}\right)\boldsymbol{\mathrm{z}}\right]} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{y}}=\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} }{\mathrm{2}\left[\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\boldsymbol{\mathrm{e}}}−\left(\boldsymbol{\mathrm{k}}+\mathrm{1}\right)\boldsymbol{\mathrm{z}}\right]}\:−\frac{\boldsymbol{\mathrm{z}}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\boldsymbol{\mathrm{e}}}}\\{\boldsymbol{\mathrm{e}}=\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} }}\end{cases} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{a}}^{\mathrm{2}} =\left(\boldsymbol{\mathrm{y}}−\frac{\boldsymbol{\mathrm{k}}}{\mathrm{2}}\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}\right)\boldsymbol{\mathrm{z}}^{\mathrm{2}} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{y}}=\frac{\boldsymbol{\mathrm{k}}}{\mathrm{2}}\boldsymbol{\mathrm{z}}+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\left(\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}\right)\boldsymbol{\mathrm{z}}^{\mathrm{2}} } \\ $$$$ \\ $$$$\boldsymbol{\mathrm{donc}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{lequati}}\mathrm{o}\boldsymbol{\mathrm{n}}\:\:\boldsymbol{\mathrm{en}}\:\boldsymbol{\mathrm{z}} \\ $$$$ \\ $$$$\:\boldsymbol{\mathrm{z}}^{\mathrm{2}} \sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}}\:+\:\boldsymbol{\mathrm{z}}\left(\frac{\boldsymbol{\mathrm{k}}+\mathrm{1}}{\mathrm{2}}\right)−\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} }{\mathrm{2}\left[\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\boldsymbol{\mathrm{e}}}\:−\left(\boldsymbol{\mathrm{k}}+\mathrm{1}\right)\boldsymbol{\mathrm{z}}\right]} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\boldsymbol{\mathrm{e}}} \\ $$$$\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{equation}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{long}}\: \\ $$$$ \\ $$$$\boldsymbol{\mathrm{othere}}\:\boldsymbol{\mathrm{methode}}: \\ $$$$−−−−−−−−− \\ $$$$..\left(\mathrm{i}\right)\Rightarrow\:\:\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\mathrm{e}}\:−\boldsymbol{\mathrm{z}}=\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right) \\ $$$$\:\boldsymbol{\mathrm{k}}+\mathrm{2}=\frac{\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\boldsymbol{\mathrm{xy}}}=\frac{\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{a}}^{\mathrm{2}} }{\boldsymbol{\mathrm{yz}}} \\ $$$$\Rightarrow\:\:\:\frac{\left.\mathrm{x}+\mathrm{y}\right)−\mathrm{c}}{\mathrm{x}}=\frac{\left.\mathrm{y}+\mathrm{z}\right)−\mathrm{a}}{\mathrm{z}} \\ $$$$\:\:\:\:\:\: \\ $$$$\Rightarrow\mathrm{a}^{\mathrm{2}} +\:\:\:\frac{\mathrm{z}\left[\left(\mathrm{x}+\mathrm{y}\right)−\mathrm{c}^{\mathrm{2}} \right]}{\mathrm{x}}=\left(\mathrm{y}+\mathrm{z}\right) \\ $$$$ \\ $$$$\frac{\left.\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)−\boldsymbol{\mathrm{c}}}{\boldsymbol{\mathrm{x}}}=\frac{\left.\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)−\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{z}}}=\frac{\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} +\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{z}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\frac{\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{z}}\right)\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{z}}+\mathrm{2}\boldsymbol{\mathrm{y}}\right)+}{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{z}}}=\frac{\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{z}}+\mathrm{2}\boldsymbol{\mathrm{y}}\right)}{\mathrm{1}}+\left(\frac{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{c}}}{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{z}}}\right)\:\: \\ $$$$ \\ $$$$\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} =\frac{\boldsymbol{\mathrm{z}}\left[\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right]}{\boldsymbol{\mathrm{x}}}+\boldsymbol{\mathrm{a}}^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{y}+\boldsymbol{\mathrm{z}}=\left(\frac{\boldsymbol{\mathrm{k}}}{\mathrm{2}}+\mathrm{1}\right)\boldsymbol{\mathrm{z}}+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} } \\ $$$$\:\:\:\:\frac{\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{a}}^{\mathrm{2}} }{\boldsymbol{\mathrm{yz}}}=\boldsymbol{\mathrm{k}}+\mathrm{2} \\ $$$$\Rightarrow\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} =\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)\boldsymbol{\mathrm{yz}}+\boldsymbol{\mathrm{a}}^{\mathrm{2}} \\ $$$$\:\:\:\:=\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)\boldsymbol{\mathrm{z}}\left(\frac{\boldsymbol{\mathrm{k}}}{\mathrm{2}}\boldsymbol{\mathrm{z}}+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} }\:\right) \\ $$$$\boldsymbol{\mathrm{donc}} \\ $$$$ \\ $$$$\left(\boldsymbol{\mathrm{z}}+\frac{\boldsymbol{\mathrm{k}}}{\mathrm{2}}\boldsymbol{\mathrm{z}}+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{3}} −\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} \:}\:\right)^{\mathrm{2}} = \\ $$$$\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\frac{\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)\boldsymbol{\mathrm{k}}}{\mathrm{2}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} +\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)\boldsymbol{\mathrm{z}}\sqrt{\boldsymbol{\mathrm{a}}−\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} }\:. \\ $$$$\:\:\:\: \\ $$$$\boldsymbol{\mathrm{soit}}\:\mathrm{apres}\:\mathrm{calculs} \\ $$$$ \\ $$$$\left(\frac{\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} −\frac{\boldsymbol{\mathrm{k}}\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)}{\mathrm{2}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} \right)−\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}}{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} =\mathrm{0} \\ $$$$\frac{\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left(\boldsymbol{\mathrm{k}}+\mathrm{2}\right)}{\mathrm{4}}=\frac{\mathrm{4}−\boldsymbol{\mathrm{k}}}{\mathrm{4}} \\ $$$$\boldsymbol{\mathrm{k}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{k}}−\mathrm{4}=\mathrm{0} \\ $$$$\:\:\bigtriangleup=\mathrm{25}\:\:\:\:\begin{cases}{\boldsymbol{\mathrm{k}}=+\mathrm{1}}\\{\boldsymbol{\mathrm{k}}=−\mathrm{4}}\end{cases} \\ $$$$\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{choisit}}\:\:\boldsymbol{\mathrm{k}}>\mathrm{0}\:\:\:\:\boldsymbol{\mathrm{k}}=\mathrm{1} \\ $$$$\mathrm{3}=\frac{\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\boldsymbol{\mathrm{xy}}}\:\:\:\:\:=\frac{\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{a}}^{\mathrm{2}} }{\boldsymbol{\mathrm{yz}}} \\ $$$$\: \\ $$$$\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\mathrm{3}\boldsymbol{\mathrm{xy}}−\boldsymbol{\mathrm{c}}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\:\left(\boldsymbol{\mathrm{x}}−\frac{\mathrm{3}}{\mathrm{2}}\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{9}}{\mathrm{4}}\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)=\mathrm{0}\:\:\left(\boldsymbol{\mathrm{ii}}\right) \\ $$$$\:\: \\ $$$$\left(\boldsymbol{\mathrm{y}}−\frac{\mathrm{3}}{\mathrm{2}}\boldsymbol{\mathrm{z}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{9}}{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)=\mathrm{0}\:\:\:\left(\boldsymbol{\mathrm{iii}}\right) \\ $$$$\boldsymbol{\mathrm{y}}=\frac{\mathrm{2}}{\mathrm{2}}\boldsymbol{\mathrm{z}}+\sqrt{\boldsymbol{\mathrm{a}}−\frac{\mathrm{3}}{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} } \\ $$$$\Rightarrow\left(\mathrm{a}−\frac{\mathrm{3}}{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} \right)=\boldsymbol{\mathrm{c}}^{\mathrm{2}} +\frac{\mathrm{9}}{\mathrm{4}}\boldsymbol{\mathrm{z}}^{\mathrm{2}} \\ $$$$\mathrm{3}\boldsymbol{\mathrm{z}}^{\mathrm{2}} =\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} \:\:\:\:\:\boldsymbol{\mathrm{z}}=\sqrt{\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{3}}} \\ $$$$\:\:\begin{cases}{\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)=\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)}\:−\sqrt{\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{3}}}\:}\\{\boldsymbol{\mathrm{xy}}\:\:\:\:\:\:=\frac{\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{3}}=\frac{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}\right)+\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{3}}−\frac{\mathrm{2}}{\mathrm{3}}\sqrt{\left.\mathrm{2}\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right)\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right.}\:−\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{3}}}\end{cases} \\ $$$$ \\ $$$$\begin{cases}{\boldsymbol{\mathrm{xy}}=\left(\frac{\mathrm{2}}{\mathrm{9}}\sqrt{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{c}}}\:\right)\left[\left(\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} }\:−\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }\:\right)\right]+\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{9}}}\\{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:=\sqrt{\frac{\mathrm{2}}{\mathrm{3}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} +\boldsymbol{\mathrm{c}}^{\mathrm{2}} \right.}\:−\sqrt{\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{c}}^{\mathrm{2}} }{\mathrm{3}}}}\end{cases} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{k}}^{\mathrm{2}} −\boldsymbol{\mathrm{sk}}+\boldsymbol{\mathrm{p}}=\mathrm{0}}\\{\boldsymbol{\mathrm{s}}=\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)\:\:\boldsymbol{\mathrm{p}}=\boldsymbol{\mathrm{xy}}}\end{cases} \\ $$$$\boldsymbol{\mathrm{k}}=\frac{\boldsymbol{\mathrm{s}}\pm\sqrt{\boldsymbol{\mathrm{s}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{p}}}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{s}}^{\mathrm{2}\:} =\mathrm{3}\boldsymbol{\mathrm{xy}}+\boldsymbol{\mathrm{c}}^{\mathrm{2}} \:\:\:\boldsymbol{\mathrm{p}}=\boldsymbol{\mathrm{xy}} \\ $$$$\Rightarrow\:\:\boldsymbol{\mathrm{k}}=\frac{\mathrm{3}\boldsymbol{\mathrm{xy}}\pm\sqrt{\mathrm{9}\boldsymbol{\mathrm{xy}}−\mathrm{4}\boldsymbol{\mathrm{xy}}}}{\mathrm{2}} \\ $$$$=\frac{\mathrm{3}\boldsymbol{\mathrm{xy}}+\sqrt{\mathrm{5}\boldsymbol{\mathrm{xy}}}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\boldsymbol{\mathrm{xy}}}\:\left[\mathrm{3}\sqrt{\boldsymbol{\mathrm{xy}}}\:+\mathrm{5}\right] \\ $$$$\begin{cases}{\sqrt{\boldsymbol{\mathrm{x}}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\sqrt{\boldsymbol{\mathrm{y}}}\:\left(\mathrm{3}\sqrt{\boldsymbol{\mathrm{xy}}}\:+\mathrm{5}\right)\:\:\:\:\:\:\right.}\\{\sqrt{\boldsymbol{\mathrm{y}}}\:=\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\boldsymbol{\mathrm{x}}}\:\left(\mathrm{3}\sqrt{\boldsymbol{\mathrm{xy}}}\:−\mathrm{5}\right)}\end{cases} \\ $$$$ \\ $$$$\sqrt{\boldsymbol{\mathrm{xy}}}\:=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\boldsymbol{\mathrm{xy}}}\:\left(\mathrm{3}\sqrt{\boldsymbol{\mathrm{xy}}}\:+\mathrm{5}\right)\left(\mathrm{3}\sqrt{\boldsymbol{\mathrm{xy}}}\:−\mathrm{5}\right) \\ $$$$\mathrm{1}=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{3}\boldsymbol{\mathrm{m}}+\mathrm{5}\right)\left(\mathrm{3}\boldsymbol{\mathrm{m}}−\mathrm{5}\right)\:\:\:\:\:\:\:\boldsymbol{\mathrm{m}}=\sqrt{\boldsymbol{\mathrm{xy}}} \\ $$$$\mathrm{9}\boldsymbol{\mathrm{m}}^{\mathrm{2}} −\mathrm{25}=\mathrm{4}\:\:\boldsymbol{\mathrm{m}}=\frac{\sqrt{\mathrm{29}}}{\mathrm{3}} \\ $$$$\:\:\:\:\sqrt{\boldsymbol{\mathrm{xy}}}\:=\frac{\sqrt{\mathrm{29}}}{\mathrm{3}}\:\:\:\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\sqrt{\boldsymbol{\mathrm{c}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{xy}}}\right. \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}=\sqrt{\boldsymbol{\mathrm{c}}^{\mathrm{2}} +\frac{\mathrm{29}}{\mathrm{3}}} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\:\:\:\:=\sqrt{\boldsymbol{\mathrm{c}}^{\mathrm{2}} +\frac{\mathrm{29}}{\mathrm{3}}}}\\{\boldsymbol{\mathrm{xy}}\:\:\:\:\:\:\:\:=\frac{\mathrm{29}}{\mathrm{9}}}\end{cases} \\ $$$$\begin{cases}{\bigtriangleup=\frac{\mathrm{3}\boldsymbol{\mathrm{c}}^{\mathrm{2}} +\mathrm{29}}{\mathrm{3}}−\frac{\mathrm{116}}{\mathrm{9}}}\\{\begin{cases}{\boldsymbol{\mathrm{x}}=\frac{\frac{\mathrm{1}}{\mathrm{3}}\left(\sqrt{\mathrm{27}\boldsymbol{\mathrm{c}}^{\mathrm{2}} −\mathrm{29}}\:−\sqrt{\mathrm{9}\boldsymbol{{c}}^{\mathrm{2}} +\mathrm{29}}\:\right)}{\mathrm{6}}}\\{\boldsymbol{\mathrm{y}}=\frac{\frac{\mathrm{1}}{\mathrm{3}}\left(\sqrt{\mathrm{27}\boldsymbol{\mathrm{c}}^{\mathrm{2}} −\mathrm{29}}\:+\sqrt{\mathrm{9}\boldsymbol{\mathrm{c}}^{\mathrm{2}} +\mathrm{29}}\:\right)}{\mathrm{6}}}\end{cases}}\end{cases} \\ $$$$\boldsymbol{{Recap}}\:\:\:\boldsymbol{{k}}=\mathrm{1}\:\: \\ $$$$\boldsymbol{{x}}=\frac{\sqrt{\mathrm{27}\boldsymbol{{c}}^{\mathrm{2}} −\mathrm{29}}\:−\sqrt{\mathrm{9}\boldsymbol{{c}}^{\mathrm{2}} +\mathrm{29}}}{\mathrm{18}} \\ $$$$\boldsymbol{{y}}=\frac{\sqrt{\mathrm{27}\boldsymbol{{c}}^{\mathrm{2}} −\mathrm{29}}\:+\sqrt{\mathrm{9}\boldsymbol{{c}}^{\mathrm{2}} +\mathrm{29}}}{\mathrm{18}} \\ $$$$\boldsymbol{{z}}=\frac{\sqrt{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{c}}^{\mathrm{2}} }}{\mathrm{3}} \\ $$$$\boldsymbol{\mathrm{pour}}\:\:\left(\:\boldsymbol{\mathrm{k}}=−\mathrm{4}\right):\:\:\:\left(\boldsymbol{\mathrm{meme}}\:\boldsymbol{\mathrm{procedure}}\right) \\ $$$$ \\ $$
Commented by mr W last updated on 09/Sep/24
$${thanks}\:{for}\:{trying}!\: \\ $$$${but}\:{you}\:{can}\:{not}\:{determine}\:{the}\:{value}\: \\ $$$${of}\:{k}\:{by}\:{yourself}.\:{k}\:{is}\:{a}\:{given}\:{constant},\: \\ $$$${for}\:{example}\:{k}=\mathrm{1}.\mathrm{5}. \\ $$
Commented by a.lgnaoui last updated on 09/Sep/24
$$\mathrm{thank}\:\mathrm{you}\: \\ $$
Answered by ajfour last updated on 08/Oct/24
$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={s} \\ $$$${kxy}+{z}^{\mathrm{2}} ={s}−{c}^{\mathrm{2}} \\ $$$${kyz}+{x}^{\mathrm{2}} ={s}−{a}^{\mathrm{2}} \\ $$$${kzx}+{y}^{\mathrm{2}} ={s}−{b}^{\mathrm{2}} \\ $$$${If}\:\:{x}={py}\:\:,\:\:{z}={qy} \\ $$$$\Rightarrow\:\:\left({kp}+{q}^{\mathrm{2}} \right){y}^{\mathrm{2}} ={s}−{c}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\left({kq}+{p}^{\mathrm{2}} \right){y}^{\mathrm{2}} ={s}−{a}^{\mathrm{2}} \\ $$$$\:\:\:\:\&\:\:\left({kpq}+\mathrm{1}\right){y}^{\mathrm{2}} ={s}−{b}^{\mathrm{2}} \\ $$$${s}=\left(\mathrm{1}+{p}^{\mathrm{2}} +{q}^{\mathrm{2}} \right){y}^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}+{kp}−{p}^{\mathrm{2}} }{\mathrm{1}+{kq}−{q}^{\mathrm{2}} }=\frac{{c}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:\:\:\:…\left({i}\right) \\ $$$$\&\:\:\frac{\mathrm{2}+{k}\left({p}+{q}\right)−\left({p}^{\mathrm{2}} +{q}^{\mathrm{2}} \right)}{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} −{kpq}}=\frac{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\:\:\:..\left({ii}\right) \\ $$$$… \\ $$