When the circle makes one complete roll on the x-axis along the positive direction of the x-axis, it moves a distance equal to its circumference, which is 10π units.
Since the circle originally touches both axes and lies in the first quadrant, its original position can be represented by the equation (x - 5)^2 + (y - 5)^2 = 25 (using the fact that the center is at (5, 5) and the radius is 5).
After one complete roll on the x-axis, the new position of the circle is obtained by adding 10π to the x-coordinate of the center, while the y-coordinate remains unchanged.
So, the equation of the circle in its new position is (x - 5 + 10π)^2 + (y - 5)^2 = 25.
Expanding this equation:
(x - 5)^2 + 20π(x - 5) + (10π)^2 + (y - 5)^2 = 25
x^2 - 10x + 25 + 20πx - 100π + 100π^2 + y^2 - 10y + 25 = 25
x^2 + y^2 + 20πx - 10y + 100π^2 = 0
Hence, the correct option is:
a. x^2 + y^2 + 20πx - 10y + 100π^2 = 0
