Website owner: James Miller
DIAMETERS OF A CONIC, CONJUGATE DIAMETERS
Def. Diameter of a conic. Any straight line which is the locus of the midpoints of a family of parallel chords.
Example. In Figure 1 chord AB is a diameter of the conic. AB is said to be conjugate to the family of parallel chords inclined at angle α.
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 .
Def. 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. Then AB and CD are conjugate diameters.
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