The polar of every point at infinity is a diameter.
The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter.
All diameters pass through a common point, the pole of the line at infinity.
All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity.
In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve.
A centre of a conic bisects every chord through it.
The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse.
The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hence also—
From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called Asymptotes (§ 59).
To construct a diameter of a conic, draw two parallel chords and join their middle points.