A line between two points of a conic is called a chord.
Every conic has infinitely many diameters and they pass through one point called
the ellipse in real
points.hyperbola in real points while
its conjugate diameter intersects this hyperbola in a pair of conjugate
imaginary points.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 |