Hyperbola 2Given the two axial vertices (V1 and V2) and a tangent line, construct the hyperbola and the point of tangency.  For a solution to exist, the given vertices must fall on opposite sides of the tangent line. This is because all tangent lines intersect the line segment joining the vertices. Construct the transverse axis, V1V2. Let A be the intersection of the axis and the line of tangency. Construct B such that AB divides V1V2 harmonically. Through B construct the line perpendicular to the axis, and let it intersect the tangent line at C. This is the point of tangency.  The figure now has both axial vertices and one other point (C) on curve. Using the construction in Hyperbola 1 as a guide, complete the construction. The completed construction is in the websketch below. Drag the given objects to confirm that the solution holds up. |