Question

The circle x² + y² − 2x − 2y + 1 = 0 is rolled along the positive direction of x-axis and makes one complete roll. Find its equation in new-position.

Answer 1

Centre of the given circle = $\left(1,1\right)$
Radius of the given circle = 1

This circle is rolled along the positive direction of the x-axis. When it makes one complete roll, its centre moves horizontally through a distance equal to its circumference, i.e 2$\mathrm{\pi }$.

Thus, the coordinates of the centre of the new circle will be $\left(1+2\mathrm{\pi },1\right)$.

Hence, the required equation of the circle is ${\left(\mathrm{x}-1-2\mathrm{\pi }\right)}^2+{\left(y-1\right)}^2=1$.