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 $z$.

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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 $z$.