1. Understanding the Problem and Setting Up the General Equation of the Circle

The problem asks for the locus of the center of a circle that satisfies two specific geometric conditions. A locus is the set of all points that satisfy a given condition or set of conditions. In this case, we are looking for the path traced by the center of such a circle.

Let the center of the circle be $(h, k)$ and its radius be $R$. The general equation of a circle with center $(h, k)$ and radius $R$ is: $(x-h)^2 + (y-k)^2 = R^2$

Our goal is to use the given conditions to form equations involving $h, k,$ and $R$. Then, we will eliminate $R$