Question

In the diagram, $M$ is a concave mirror and $A$ is a point on its principal axis. If an object $O$ is kept at $A$, the image is formed at $A$ itself. $B$ is the center of $AP$. What is the point $B$ called?

Answer 1

Object and image position coincide in the concave mirror only if the object is placed at the center of curvature.

So, $PA$ is radius of curvature.
Since radius of curvature is twice the focal length, $B$ is the focus.