Hyperbola 5Given both asymptotes (asymptote1 and asymptote2) and one point (P) on curve, construct the hyperbola.  The asymptotes intersect at point O, the center. Through point P construct the line parallel to asymptote1. Let it intersect asymptote2 at A. Dilate A with respect to center O, using scale factor 2, and let the image point be B. Produce line BP to point C, on asymptote1.  Point P is the midpoint of line segment BC, which is tangent to the hyperbola at P. Construct the circle with center O and radius equal to the geometric mean of OB and OC. Let the circle intersect ray OB at D and ray OC at E. Let point V1 be the midpoint of DE. Line segment DE is tangent to the curve at V1, which is an axial vertex. Dilate V1 with respect to O, using scale factor –1. Let the image be V2, the other vertex. Rays, rather than lines, are used here to ensure that V1 falls in the same quadrant of the asymptote intersection as P. (See Conica (II, 3) and (III, 43).  The sketch now has both axial vertices and one other point on curve. Use Hyperbola 1 as a guide for completing the construction. The websketch below has the completed construction. Drag the given objects to confirm it. |