A longbow curve is defined as follows. Fix a circle in the xy-plane with radius a and center (0, a), and fix the horizontal line y = 2a. Each nonhorizontal line through the origin intersects the circle in a point A and the line in a point B. Let P be the midpoint of the segment from A to B. The set of points P obtained in this way is the longbow curve. (When you sketch it, you will see why it has this name.) By letting θ be the angle that a line through the origin forms with the positive x-axis, find a parametrization of the longbow curve in terms of the parameter θ.
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