The Concentric Circle construction of the hyperbola

Back to Geometry homepage

The points \(F_1\) and \(F_2\) are the focal points of the hyperbola, which determine the major axis (the line \(F_1 F_2\)) and the minor axis (the perpendicular bisector of \(F_1 F_2\)). The two blue circles have centre \(O\) (the midpoint of \(F_1 F_2\)).
Given any ray from the centre, let \(X\) be the point where it intersects the line tangent to the inner circle and perpendicular to the major axis \(F_1 F_2\), and let \(Y\) be the point where the ray intersects the outer circle. Now draw a line tangent to the outer circle through the point \(Y\) (coloured green in the diagram), and let \(Z\) be the point where this line intersects the major axis. Draw a line through \(Z\) parallel to the minor axis and a line through \(X\) parallel to the major axis (these lines are coloured red in the diagram). The concentric circle construction says that these two lines will intersect at a point \(A\) on the hyperbola.
You can click and drag the points \(O\), \(P\) and \(Q\) to move the centre, the major/minor axes and the radii of the two circles. You can pause the animation and click and drag the point \(Y\) to change the direction of the ray from the centre.