Hyperbola 1Given the two axial vertices (V1 and V2) and one other point (P) on curve, construct the hyperbola.  Construct transverse axis V1V2. Let center O be the midpoint of V1V2, and construct the circle with diameter V1V2. Let line PA be perpendicular to the axis, meeting it at A. Construct the circle with diameter OA. Let point B be one of the intersections of the two circles, so that BA is tangent to circle O. Let line PC be parallel to the axis, meeting OB at C. Let line k pass through C, perpendicular to the axis. The distance from O to line k is equal to half the length of the conjugate axis.  Construct a point D on circle O, and let DE be perpendicular to OD, meeting V1V2 at E. Let F be the intersection of OD and line k. Construct G, where FG is parallel to the axis, and EG is perpendicular to it. Construct the locus of G as D travels its path. This is the required hyperbola.  Rotate line k by 90° about O, and label its image kʹ. Construct V2H perpendicular to V1V2, meeting kʹ at H. Line OH is one of the asymptotes. Let the circle through the vertices intersect OH at J. The line through J and perpendicular to V1V2 is one of the directrices. Construct the circle with center O and passing through H. This circle intersects V1V2 at the foci.  Construct the conjugate axis through O, perpendicular to the transverse axis. Complete the figure be reflecting the asymptote and the directrix across the conjugate axis.  In the interactive sketch below drag the given objects to confirm that the solution holds up. |