Given equation of the circle is x² + y² - 2x - 2y + 1 = 0.

We know that the equation of the circle with centre (p,q) and having radius ‘r’ is given by:

⇒ (x - p)² + (y - q)² = r² ..... - (1)

Now,

⇒ x² + y² - 2x - 2y + 1 = 0

⇒ (x² - 2x + 1) + (y² - 2y + 1) = 1

⇒ (x - 1)² + (y - 1)² = (1)² ..... (2)

Comparing (2) with (1), we get

⇒ Centre = (1,1) and radius = 1

It is told that the circle is rolled along the positive direction of x - axis and makes one complete roll.

We know that the complete roll of a circle covers the distance 2r, where r is the radius of the circle.

The centre of the circle as moves 2r in the positive direction of xthe - axis.

Let the d be the distance moved by the centre on completion of one roll.

⇒

⇒ d = 2

The new position of the centre is (1 + d,1)

⇒ Centre = (1 + 2π, 1)

We have circle with centre (1 + 2π, 1) and having radius 1 units.

We know that the equation of the circle with centre (p, q) and having radius ‘r’ is given by:

⇒ (x - p)² + (y - q)² = r²

Now we substitute the corresponding values in the equation:

⇒ (x - (1 + 2π))² + (y - 1)² = 1²

⇒ (x - 1 - 2π)² + (y - 1)² = 1

⇒ x² - (2 + 4π)x + (1 + 2π)² + y² - 2y + 1 = 1

⇒ x² + y² - (2 + 4π)x - 2y - (1 + 2π)² = 0

∴The equations of the circles is x² + y² - (2 + 4π)x - 2y - (1 + 2π)² = 0.