$4 \cdot 126$ Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=a \\
x^{2}+y^{2}+z^{2}=b^{2} \\
x y=z^{2}
\end{array}\right.
$$

where $a, b$ are known numbers. Also, indicate the conditions that $a, b$ must satisfy for $x, y, z$ to be distinct positive numbers.
[Solution] From (1) $)^{2}-$ (2) we get $x y+y z+z x=\frac{1}{2}\left(a^{2}-b^{2}\right)$, i.e., $\approx a=\frac{1}{2}\left(a^{2}-b^{2}\right)$.
Therefore, the solution is
$$
\left\{\begin{array}{l}
x=\frac{1}{4 a}\left(a^{2}+b^{2} \pm \sqrt{10 a^{2} b^{2}-3 a^{4}-3 b^{4}}\right), \\
y=\frac{1}{4 a}\left(a^{2}+b^{2} \mp \sqrt{10 a^{2} b^{2}-3 a^{4}-3 b^{4}}\right), \\
z=\frac{1}{2 a}\left(a^{2}-b^{2}\right) .
\end{array}\right.
$$

When and only when $|b|<a<\sqrt{3}|b|$, $x$, $y$, and $z$ are distinct positive numbers.