Given an ellipse, and a general means of finding the intersection of the ellipse and a line or circle, locate the foci of the ellipse. This is an extended ruler and compass construction. Pick two points on the ellipse and draw a line thru them. Pick another point on the ellipse and draw a line parallel to the first. Repeat previous operation until 2nd line intersects ellipse twice. Bisect both line segments. Draw a line N thru the midpoints. N goes thru the center of the Ellipse and intersects it twice. Find the point midway between the two intersections of N with the ellipse. That is the center C of the ellipse. Pick a new point on the ellipse and draw a circle about C thru the point. Repeat the previous step until the circle intersects the ellipse 4 times. If that loop goes more than 5 trials the ellipse is a circle and the center is also the foci. Draw a line thru C parallel to a chord connecting two of the adjacent intersections of the circle and ellipse. This line is the first axis. Draw a line thru C perpendicular to the first axis. This is the second axis. Draw a circle about the end of the shorter axis with radius which is half the length of the longer axis. This circle intersects the long axis at the two foci.
This construction works for a hyperbola as well as an ellipse except one of the axes fails to intersect the hyperbola. Following a suggestion by Lee Corbin we construct the asymptotes as follows. Adopt a Cartesian coordinate system such that the vertices of the hyperbola are at <−1, 0> and <1, 0>. Consider the hyperbola x2 − y2 = 1. Its asymptotes are y = ±x. The point <5/3, 4/3> lies on the new hyperbola and we can construct this point. The point <5/3, 5/3> lies on the asymptote of the new hyperbola. Construct the line N perpendicular to the X-axis and thru <5/3, 4/3>. N intersects the old hyperbola at a point 4/5 of the way to where N intersects the asymptote for the old hyperbola. We can now construct the asymptotes of the original hyperbola.
For the foci: draw a line thru <1, 0> perpendicular to the X-axis. It will intersect the asymptote at Q.