Example 3. For the equation $t^{2} + z t + z i = 0$ with $z$ and $i$ as coefficients, it always has a real root $\alpha$. Find the locus equation of point $z$.

保持源文本的换行和格式如下：

Example 3. For the equation $t^{2} + z t + z i$ $=0$ with $z$ and $i$ as coefficients, it always has a real root $\alpha$. Find the locus equation of point $z$.
Let $z=x+y i$, we need to find the relationship between $x$ and $y$. Substituting $\lambda a^{2}+\alpha z+z i=0$, and applying the condition for equality of complex numbers, we get $\boldsymbol{a}^{2}+\boldsymbol{\alpha} x-y=0, a y+x=0$. If $y=0$, then $x=0$; assuming $y \neq 0$, then $a=-\frac{x}{y}$. Substituting into the first equation and rearranging, we get $x^{2}(1-y)=y^{3}, \quad(0,0)$ also satisfies the equation, which is the desired result.
(Author's affiliation: Shandong Provincial County No.1 High School)