Animated Solution for Mathematics - Circles: The centres of a set of circles, each of radius 3, lie on the circle $x^2+y^2=25$. The locus of any point in the set is

Let the fixed circle be denoted by $S_1$, defined by the equation $x^2+y^2=25$. This circle is centered at the origin $O(0,0)$ and has a radius $R=5$.

Let $C(h,k)$ be the center of a moving circle $S_2$ with radius $r=3$. Since the center $C$ is constrained to lie on the circumference of $S_1$, the distance from the origin to $C$ is fixed:

Consider an arbitrary point $P(x,y)$ located on the circumference of the moving circle $S_2$. By the definition of a circle, the distance between the center $C$ and any point $P$ on its boundary is equal to the radius:

We now consider the triangle $△OCP$ formed by the origin, the center of the moving circle, and the point $P$. According to the triangle inequality, the distance $OP$ is bounded by the sum and the difference of the lengths of the other two sides:

Substituting the known values $OC=5$ and $CP=3$ into the inequality, we obtain:

Since the distance $OP$ is defined as $\sqrt{x^2+y^2}$, we substitute this into our inequality:

To determine the locus, we square all parts of the inequality to eliminate the square root:

This yields the final expression for the locus:

The Annular Region

The locus of point $P$ is an annular region (a ring) centered at the origin. This region is bounded by two concentric circles: an inner circle with radius $2$ and an outer circle with radius $8$.

Any point $P$ on any of the moving circles will always be trapped within this shaded region. This result demonstrates the power of geometric visualization in solving complex locus problems.