I want to cut a border exactly like the stroke. So i applied offset. But the offset's end is not as pointy as the stroke.it ends up as flat/smooth.
This a workaround which doesn't involve making an offset.
I assume you made the curved shape as a stroke with Width Profile 1. If not, then I suggest you create it that way.
Select the curved shape, copy it, and Paste in Back
Set the stroke colour to white, increase the stroke width
Use the Curvature Tool to click and drag the ends towards the edges of the circle.
Do Object > Expand Appearance
Now select both the black circle and white inner shape, open the Pathfinder and hit Minus Front to cut it out.
To have the result that you want to achieve, it is more preferable that you use the stroke setting, instead of using offset. You will find under the stroke setting (I have it in german) how much the max of a corner should be. If you get some very long corners, then you must adjust the corners of the pink stroke with the pen tool.