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   DIAMETERS OF A CONIC, CONJUGATE DIAMETERS

Diameter of a conic. Any straight line which is the locus of the midpoints of a family of parallel chords. See Figure 1.

Equation of the diameter of a conic. Given: conic

                        ax² + 2hxy + by² + 2gx + 2fy + c = 0 . 

The equation for the diameter conjugate to the chords inclined at an angle α with respect to the positive x axis is

                        (ax + hy + g) cos α + (hx + by + f) sin α = 0 .

Conjugate diameters. Let Q be a system of parallel chords cutting an ellipse G in direction α and let AB be the diameter conjugate to them. See Figure 1. Let S be a system of parallel chords cutting the ellipse in direction β, parallel to AB, and let CD be the diameter conjugate to them. See Figure 2. AB and CD are conjugate diameters.