Hyperbola 3Given the two foci (F1 and F2) and a point (P) on curve, construct the hyperbola.  The line joining F1 and F2 is the transverse axis. Construct center O, the midpoint of F1F2. Construct the circle with center P, passing through F2. Construct ray PF1 and let it intersect circle P at A. A ray is used here because A must be the intersection point nearest F1. Let B be the midpoint of AF1, and construct the circle with diameter AF1.  The distance AF1 is the difference of the distances from P to the two foci, so it must also be the distance between the axial vertices. Translate circle B by vector BO. Let the image intersect the axis at vertices V1 and V2.  At this point all the parts are in place to use the construction from Hyperbola 1, but it might be better to make use of these foci. Construct the circle with center O, passing through the foci. Construct the line through V2, perpendicular to the transverse axis. Let it intersect the circle through the foci at C and D. Lines OC and OD are the asymptotes. Construct the line through C, parallel to the axis. Rotate this line 90° about the center, and label the image k.  The distance from O to line k is equal to one-half the conjugate axis, so it will be useful for the locus construction. Construct line OE, where E is on the circle having diameter V1V2. Produce OE to meet line k at G. Construct EH perpendicular to OE, intersecting the transverse axis at H. Construct point J such that GJ is parallel to the transverse axis and HJ is perpendicular to it. Construct the locus of J as E travels its circle path.  The directrices intersect the asymptotes on this remaining circle O with diameter V1V2, and the conjugate axis is constructed through O, perpendicular to the transverse axis.  The completed construction is in the websketch below. Drag the given objects to confirm that the solution holds up. |