Question

Given $y^2+z^2=ayz,z^2+x^2=bxz,x^2+y^2=cxy$ express $\frac{y^2}{xz}+\frac{xz}{y^2}$ in terms of $a,b,c.$

Answer 1

Multiplying $1st$ and $3rd$ relations, we get
$y^2{x}^2+y^4+z^2{x}^2+z^2{y}^2={acy}^{2}xz$
or $(y^2{x}^2+y^3{z}^2)+(y^4+z^2{x}^2)={acy}^{2}xz$
Dividing by $y^{2}xz$, we get
$\frac{x^2+z^2}{xz}+\frac{y^2}{xz}+\frac{xz}{y^2}=ac$
Substituting $\frac{x^2+z^2}{xz}=b$ from $2nd$ relation in $\left(1\right)$, We get
$\frac{y^2}{zx}+\frac{zx}{y^2}=ac-b$