1. **Problem Statement:** Show that the evolute of the circle given by the equation $$x^2 + y^2 = a^2$$ reduces to a single point and explain why. 2. **Recall the definition of evolute:** The evolute of a curve is the locus of all its centers of curvature. For a curve parameterized by arc length or a parameter $t$, the evolute can be found by locating the centers of the osculating circles at each point. 3. **Circle properties:** For a circle of radius $a$, every point on the circle has the same radius of curvature $\rho = a$ because the circle is perfectly symmetric. 4. **Center of curvature for a circle:** The center of curvature at any point on the circle is the center of the circle itself, since the osculating circle at any point on a circle is the circle itself. 5. **Equation of the circle:** The circle is centered at the origin $(0,0)$ with radius $a$. 6. **Evolute calculation:** Since the radius of curvature is constant and the center of curvature is always at $(0,0)$, the evolute is the single point at the origin. 7. **Conclusion:** The evolute of the circle $$x^2 + y^2 = a^2$$ reduces to the point $(0,0)$ because the center of curvature does not change as we move along the circle; it is always the center of the circle. This shows that the evolute of a circle is a single point, the center of the circle.