Parabola 7Given the directrix and two points (P1 and P2) on the curve, construct a parabola.  Through each of the given points construct a line perpendicular to the directrix and construct the point where each line intersects it. Use these intersection points to construct circles centered on the given points and tangent to the directrix. Construct the intersections of the circles. These are foci, as each of the given points on curve will be equidistant from a focus and the directrix.  Now having the focus and directrix, the parabola, axis, and axial vertex can be constructed for each of the two solutions. If the given points P1 and P2 are on opposite sides of the directrix, or if the distance between them exceeds the sum of their distances from the directrix, there are no circle intersections, hence, no solutions. Under certain special conditions (tangent circles) there can be a single solution. Generally there are two solutions if there are any at all. |