The polar line of an infinitely distant point is called a diameter, and the pole of the infinitely distant line is called the center, of the conic.

108. From the harmonic properties of poles and polars,

The center bisects all chords through it (§ 39).

Every diameter passes through the center.

All chords through the same point at infinity (that is, each of a set of parallel chords) are bisected by the diameter which is the polar of that infinitely distant point.

109.  Conjugate diameters. We have already defined conjugate lines as lines which pass each through the pole of the other (§ 100).

Any diameter bisects all chords parallel to its conjugate.

The tangents at the extremities of any diameter are parallel, and parallel to the conjugate diameter.

Diameters parallel to the sides of a circumscribed parallelogram are conjugate.

All these theorems are easy exercises for the student.

110.  Classification of conics. Conics are classified according to their relation to the infinitely distant line.  If a conic has two points in common with the line at infinity, it is called a hyperbola; if it has no point in common with the infinitely distant line, it is called an ellipse; if it is tangent to the line at infinity, it is called a parabola.

111. In a hyperbola the center is outside the curve (§ 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center.  As previously noted, these two tangents are called the asymptotes of the curve.  The ellipse and the parabola have no asymptotes.

112. The center of the parabola is at infinity, and therefore all its diameters are parallel, for the pole of a tangent line is the point of contact.

The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords.

The center of an ellipse is within the curve.

[Figure 28]

FIG. 28

113.  Theorems concerning asymptotes. We derived as a consequence of the theorem of Brianchon (§ 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point.  Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in A and B (Fig. 28).  If, then, O is the intersection of the asymptotes,—­and therefore the center of the curve,—­ then the triangle OAB is circumscribed about the curve.  By the theorem just quoted, the line through A parallel to OB, the line through B parallel to OA, and the line OP through the point of contact of the tangent AB all meet in a point C.  But OACB is a parallelogram, and PA = PB.  Therefore