As a general rule of thumb, any prograde or retrograde burn establishes a new orbit tangent to the old orbit at the point of the burn. A prograde burn increases the length of the major axis, while a retrograde burn reduces it. To demonstrate, I set up a situation similar to yours in Kerbal Space Program, and had it draw the results of two proposed maneuvers. In both, the solid blue-green line is the initial orbit, the dotted brown line is the the result of the maneuver, and the star shape is the point of the burn.
A prograde burn at your proposed "point A" extends the major axis, raising both the periapsis and apoapsis, and rotating the major axis in the prograde direction.
A prograde burn on the other side of the orbit's major axis also raises the periapsis and apoapsis, but rotates the major axis in the other direction.