If a circle of radius 5 is made to roll along the x-axis, what is the equation for the path of the center of the circle?

Answer 1

By definition of a circle, the centre is always at the same distance (=radius) from the circumference, or more precisely, any tangent to the circle.
If the circle of radius 5 rolls above the x-axis, the locus of the centre is therefore y=5. If it rolls under the x-axis, the locus of the centre is y=-5.