Question 6
In a vertical plane we consider the points $A$ and $B$ located on a horizontal line, and the semicircumference of ends $A, B$ located in the lower half plane. A segment of length $a$, equal to the diameter of the semicircle, moves in such a way that it always contains the point $A$, and that one of its ends runs through the given semicircle. Determine the value of the cosine of the angle that that segment must form with the horizontal line, so that its midpoint is as low as possible.