>The problem :
>In a given circle to draw a diameter such that it shall
>subtend a given angle at a given point. ( I placed the point
>outside of the circle )

Have a look at

http://www.cut-the-knot.org/Curriculum/Geometry/Inversion2.shtml

The page is about inversion with negative power.

All circles that pass through a given point A and two diamterically opposite points on a given circle, always pass through another point A'. You can get A' by inverting A in the given circle and then reflecting the result in the its center.

The circle you are looking for will pass through both A and A'. On the other hand you can find its radius or diameter from the condition that an inscribed angle subtends a chord of a given length (= the diameter of the given circle.)

You must be able to construct a circle of the required size through two available points.