Conjugated diameters of a conic

A line between two points of a conic is called a chord.

Midpoints of parallel chords of a conic are collinear points and the line connecting them is called a diameter of a conic. We say this diameter is conjugate to this direction of parallel chords. Tangent lines that pass through real intersections of a diameter and a conic are always parallel and have the same direction as the chords conjugate to that diameter.

Every conic has infinitely many diameters and they pass through one point called the center of the conic. Parabola's center is a point at infinity that lies on its axis.

Look at all chords parallel to one diameter. Midpoints of this chords lie on another diameter that has the same direction as the chords conjugate to the first diameter. Such two diameters are called conjugate diameters.
Two diameters are conjugate if each bisects the chords parallel to the other diameter.

The following is important:

  • Both conjugate diameters intersect an ellipse in real points.
  • One diameter intersects a hyperbola in real points while its conjugate diameter intersects this hyperbola in a pair of conjugate imaginary points.
  • All diameters of a parabola are parallel to its axis and the line at infinity is conjugate to all of them.

    Since the tangents at the intersections of a diameter and a conic are parallel and every diameter of an ellipse intersects it in real points, the following holds for conjugate diameters of an ellipe:
    Two diameters of an ellipse are conjugate if the tangents at intersections of one diameter and this ellipse are parallel to the other diameter.

    Generally, conjugate diameters of an ellipse or a hyperbola are not perpendicular. However, for these conics exists exactly one pair of perpendicular conjugate diameters. 
    Pair of perpendicular conjugate diameters of an ellipse or a hyperbola are called axes of that conic.

    Lines that carry the axes of an ellipse or a hyperbola are axes of symmetry of these curves.  Parabola has only one axis of symmetry.
    Intersection points of a conic and its axes are called vertices of a conic.
    Ellipse has four vertices, hyperbola two and parabola one.

    A circle, a special case of an ellipse, has some interesting properties. We recall from elementary school that a circle has infinitely many axes of symmetry (it is symmetrical with respect to any line passing through its center) and that in every point on a circle, the tangent at that point is perpendicular to the diameter that passes through that point. Therefore, the following holds:
    Every pair of perpendicular diameters of a circle are conjugate.

    Figure 9

    Figure 10

    Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb