\[2r\]

is a circle radius \[r\]

as shown. P is a fixed point on the circumference of the smaller circle. The smaller circle starts to roll anticlockwise around the interior of the larger circle. While the small circle rolls anticlockwise, it rotates clockwise about its own centre. In fact if the small circle rolls once about the the large circle, it rolls a length

\[L=2 \pi (2r)=4 \pi r\]

so will turn about an angle \[\frac{4 \pi r}{r}= 4 \pi\]

about its centre. In fact the small circle turns about its own centre through twice the angle that it turns about the centre of the large circle.The angle OCP is isosceles and taking the centre of the large circle as the origin, the coordinates of P are

\[(rcos \alpha + r cos \alpha , rsin \alpha - r sin \alpha )=(2r cos \alpha , 0)\]

.Hence the

\[y\]

coordinate of P is constant and the point P remains on the \[x\]

axis.