Example 6.1.1 Let $k$ be a positive real number. Find the minimum value of the following expression
$$k\left(x^{2}+y^{2}\right)+z^{2}$$

where $x, y, z>0$, and satisfy $x y+y z+z x=1$
Solution: We decompose $k=l+(k-l)(0 \leq l \leq k)$, and apply the AM-GM inequality, obtaining the following inequalities:
$$\begin{array}{l}
l x^{2}+l y^{2} \geq 2 l x y \\
(k-l) x^{2}+\frac{1}{2} z^{2} \geq \sqrt{2(k-l)} x z \\
(k-l) y^{2}+\frac{1}{2} z^{2} \geq \sqrt{2(k-l)} y z
\end{array}$$

Combining these results, we get
$$k\left(x^{2}+y^{2}\right)+z^{2} \geq 2 l x y+\sqrt{2(k-l)}(x z+y z)$$

Given the condition $x y+y z+z x=1$, we need to find a number $l$ such that $2 l=\sqrt{2(k-l)}$. Simple calculations yield $l=\frac{-1+\sqrt{1+8 k}}{4}$, so our final result is
$$k\left(x^{2}+y^{2}\right)+z^{2} \geq \frac{-1+\sqrt{1+8 k}}{2}$$

Note: The following more general problem can be solved using the same method.

Suppose $x, y, z>0$, and satisfy $x y+y z+z x=1, k, l$ are two positive constants. Then the minimum value of the following expression
$$k x^{2}+l y^{2}+z^{2}$$

is $2 t_{0}$, where $t_{0}$ is the unique positive root of the following equation.
$$2 t^{3}+(k+l+1) t-k l=0$$

Using the same method, we will solve some related problems involving intermediate variables.