Ellipse 3Given the two foci (F1 and F2) and a point (P) on curve, construct the ellipse.  These conditions are very forgiving. For a solution to exist, point P cannot fall on line segment F1F2, but anywhere else in the plane is okay. Begin with the circle centered on P and passing through F2. Let line PF1 intersect the circle at point A, where P is between F1 and A. Marking F1 as center, dilate points A and F2, by scale factor 1/2, and let their images be B and C, respectively. Construct the circle with center B and diameter F1A.  Notice now that F1A = PF1 + PF2. This is the length of the major axis of the required ellipse. Point C is the center of the ellipse. Translate circle B by vector BC. Let the image circle intersect line F1F2 at the major vertices, V1 and V2. Unlike certain other challenges, this one leaves no question as to which axis V1V2 is. The foci fall on this axis, so it can only be the major axis. The construction is now left with two corresponding vertices and a point on the curve. Use the challenge Ellipse 1 as a guide for completion of the construction.  In the interactive sketch below drag the given objects to confirm that the solution holds up. |