When I try to add a point to the curve bezier ( by selecting one of the two endpoints, then holding Ctrl and clicking LMB). Sometimes Blender add a point other times Blender duplicate the same point. The photo shows the duplicate point by selecting one of the two endpoints then holding Ctrl and cliking LMB. Why?
You may be unintentionally creating a new disconnected subcurve of the entire curve object. This happens when the active vertex is not an endpoint of the curve. Curve objects can have many disconnected subcurves. Flexibility and Complexity.
First select an end point of the curve. Next control-click the location where you want the new point to be. The new point can be moved immediately after.
Optionally Click two points and subdivide to add a new point if that suits your needs better. The new point will be in the current curve. Subdivide is on the menu or keystroke w.