<s>[INST] value of expression having variables $p,q,r,x,y,z$ 
If $p,q,r,x,y,z$ are non zero real number such that
$px+qy+rz+\sqrt{(p^2+q^2+r^2)(x^2+y^2+z^2)}=0$
Then $\displaystyle \frac{py}{qx}+\frac{qz}{ry}+\frac{rx}{pz}$ is

what  try
$(px+qy+rz)^2=(p^2+q^2+r^2)(x^2+y^2+z^2)$
$p^2x^2+q^2y^2+r^2z^2+2pqxy+2qryz+2prxz=p^2x^2+p^2y^2+p^2z^2+q^2x^2+q^2y^2+q^2z^2+r^2x^2+r^2y^2+r^2z^2$
$2pqxy+2qryz+2prxz=p^2y^2+p^2z^2+q^2x^2+q^2z^2+r^2x^2+r^2y^2$
How do i solve it Help me please
 [/INST] By C-S $$0=px+qy+rz+\sqrt{(p^2+q^2+r^2)(x^2+y^2+z^2)}\geq px+qy+rz+|px+qy+rz|$$ and since
$$px+qy+rz+|px+qy+rz|\geq0,$$ we obtain $$px+qy+rz+|px+qy+rz|=0,$$ which gives
$$px+qy+rz\leq0.$$
Also, the equality occurs for $$(x,y,z)||(p,q,r),$$ which says that there is $k<0$, for which
$$(p,q,r)=k(x,y,z).$$
Thus, $$\sum_{cyc}\frac{py}{qx}=\sum_{cyc}\frac{kxy}{kyx}=3.$$
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