Example 3 Given that a curve is the locus of points whose distances to two fixed points $O(0,0), A(3,0)$ are in the ratio $\frac{1}{2}$, find the equation of this curve.
Solution 1 From the given conditions, using property 4, we know that $a=0, b=3, m=1, n=2$. Or using property 5, we know that $a=3$, $c=0, n=1, m=2$, from which we can derive the equation of the curve as $(x+1)^{2}+y^{2}=4$.

Solution 2 From the given conditions and the definition of the trajectory of a circle, and knowing that the curve sought is a circle, we can infer that the center of the circle lies on the line $O A$, and the two intersection points of the circle with the line $O A$ are the endpoints of the diameter. Using plane geometry, the moving point $P$ that satisfies $\frac{|P O|}{|P A|}=\frac{1}{2}$ is $P(1,0), Q(-3,0)$, thus the equation sought is
$$
(x-1)(x+3)+(y-0)(y-0)=0 \text {, i.e., }(x+1)^{2}+y^{2}=4 \text {. }
$$