Example 1 Given points $A(1,-1), B(-4,5)$, point $C$ is on the line $AB$, and $|AC|=3|AB|$, find the coordinates of point $C$.
Solution: From $|A C|=3|A B|$, we know there are two possible positions for $C$.
When $C$ is on the extension of $B A$, according to the problem, point $C$ divides $A B$ in the ratio $\lambda=-\frac{3}{4}$, then the coordinates of point $C$ are $x=16, y=-19$, i.e., $C(16,-19)$.

When point $C$ is on the extension of $A B$, according to the problem, point $B$ divides the segment $A C$ in the ratio $\lambda=\frac{1}{2}$, then the coordinates $(x, y)$ of point $C$ satisfy
$-4=\frac{1+\frac{1}{2} x}{1+\frac{1}{2}}, 5=\frac{-1+\frac{1}{2} y}{1+\frac{1}{2}}$, solving these gives $x=-14, y=17$. Therefore, the coordinates of point $C$ are $(-14,17)$.
In conclusion, the coordinates of point $C$ are $(16,-19)$ or $(-14,17)$.