To find the centre of a conic, draw two diameters; their intersection will be the centre.
§ 70. Conjugate Diameters.—A polar-triangle with one vertex at the centre will have the opposite side at infinity. The other two sides pass through the centre, and are called conjugate diameters, each being the polar of the point at infinity on the other.
Of two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter.
Further—
Every parallelogram inscribed in a conic has its sides parallel to two conjugate diameters; and
Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters.
This will be seen by considering the parallelogram in the first case as an inscribed four-point, in the other as a circumscribed four-side, and determining in each case the corresponding polar-triangle. The first may also be enunciated thus—
The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters.
§ 71. If every diameter is perpendicular to its conjugate the conic is a circle.
For the lines which join the ends of a diameter to any point on the curve include a right angle.