(Example 5 Write a sufficient but not necessary condition and a necessary but not sufficient condition for the quadratic equation with real coefficients $x^{2}-2 a x+a=0$ to have two distinct positive real roots.

)
Solve The necessary and sufficient condition for the real-coefficient quadratic equation $x^{2}-2 a x+a=0$ to have two distinct positive real roots is
$$
\left\{\begin{array}{l}
\Delta=(-2 a)^{2}-4 a>0, \\
x_{1}+x_{2}=2 a>0, \\
x_{1} \cdot x_{2}=a>0,
\end{array}\right.
$$

Solving this, we get $a>1$.
Therefore, $a>2$ is a sufficient but not necessary condition for the real-coefficient quadratic equation $x^{2}-2 a x+a=0$ to have two distinct positive real roots;
$a \geqslant 1$ is a necessary but not sufficient condition for the real-coefficient quadratic equation $x^{2}-2 a x+a=0$ to have two distinct positive real roots. Key to solving the problem Generally, the key to finding a sufficient condition or a necessary condition for a conclusion to hold lies in first finding the necessary and sufficient condition for the conclusion to hold, then narrowing the range to get the sufficient condition; expanding the range to get the necessary condition.