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I have two points say A (lat,lon ) and B(lat,lon) and I want to draw a perpendicular bisector line to AB such that any point (C) on the perpendicular line satisfies AC=BC.

All I want is a function which takes A (lat,lon) B (lat,lon) and distance from the mid point of A and B and returns lat,lon (C) such AB = AC.

The end goal is to draw various circle with AB as chord.

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This is a part of the "median line" problem of maritime boundaries. I consider the solution of this problem using the azimuthal equidistant projection in Section 14 of Geodesics on an ellipsoid of revolution. Implementation of the general azimuthal equidistant projection is given by GeographicLib (for C++ and MATLAB). There are implementations of basic geodesic functions in various other languages too (JavaScript, python, etc.); and implementing the azimuthal equidistant projection is trivial once you have these.

All of this is for an ellipsoid, of course. For a sphere, you can presumably take some short cuts. However, I would be inclined just to implement the general ellipsoid solution and, if necessary, set the flattening to zero to recover the spherical result.

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