Example 8 Given
$$
\begin{array}{l}
\frac{x}{m}+\frac{y}{n}+\frac{z}{p}=1, \\
\frac{m}{x}+\frac{n}{y}+\frac{p}{z}=0,
\end{array}
$$

Calculate the value of $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}+\frac{z^{2}}{p^{2}}$.
(1996, Tianjin City Junior High School Mathematics Competition)
Solving: Squaring (1) yields
$$
\left(\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}+\frac{z^{2}}{p^{2}}\right)+2\left(\frac{x y}{m n}+\frac{y z}{n p}+\frac{z x}{m p}\right)=1 .
$$

From (2), removing the denominators gives
$$
x y p+y z m+z x n=0 \text {. }
$$

That is, $\frac{x y}{m n}+\frac{y z}{n p}+\frac{z x}{m p}=0$.
Therefore, $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}+\frac{z^{2}}{p^{2}}=1$.