<s>[INST] minimal value of $x^2+2y^2+5z^2$ with constraint. $x,y,z>0$, and $xy+yz+zx=1$.
I need to find the minimum value of 
$x^2+2y^2+5z^2$ 
In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers with $a,b,c>0 $
 [/INST] Using AM-GM, we have:
$$4x^2+9y^2 \ge 12xy$$
$$3y^2+12z^2 \ge 12yz$$
$$18z^2+2x^2\ge 12zx$$
Combining and using the constraint, 
$$6x^2+12y^2+30z^2 \ge 12(xy+yz+zx) \implies x^2+2y^2+5z^2 \ge 2$$
Equality is when $x:y:z = 3:2:1$ and satisfying the constraint, i.e. when $x = \frac3{\sqrt{11}}, y = \frac2{\sqrt{11}}, z = \frac1{\sqrt{11}}$.
In general for $ax^2+by^2+cz^2$, you can construct similar inequalities using unknown coefficients to start with, which you solve to keep the equality condition and match LHS / RHS.
</s>