A geometry problem by Pankhuri Agarwal

Consider a circle in the $XY$ plane with diameter $1$ , passing through the origin $O$ and through a point $A=(1,0)$ . For any point $B$ on the circle, let $C$ be the point of intersection of the line $OB$ with the vertical line through $A$ . If $M$ is the point on the line $OBC$ such that $OM$ and $BC$ are of equal length, then the locus of point $M$ as $B$ varies is given by the equation $\text{\_\_\_\_\_\_\_\_\_\_}$ .