[Figure 26]

FIG. 26

96.  Definition of the polar line of a point. Consider the quadrilateral K, L, M, N inscribed in the conic (Fig. 26).  It determines the four harmonic points A, B, C, D which project from N in to the four harmonic points M, B, K, O.  Now the tangents at K and M meet in P, a point on the line AB.  The line AB is thus determined entirely by the point O.  For if we draw any line through it, meeting the conic in K and M, and construct the harmonic conjugate B of O with respect to K and M, and also the two tangents at K and M which meet in the point P, then BP is the line in question.  It thus appears that the line LON may be any line whatever through O; and since D, L, O, N are four harmonic points, we may describe the line AB as the locus of points which are harmonic conjugates of O with respect to the two points where any line through O meets the curve.

97. Furthermore, since the tangents at L and N meet on this same line, it appears as the locus of intersections of pairs of tangents drawn at the extremities of chords through O.

98. This important line, which is completely determined by the point O, is called the polar of O with respect to the conic; and the point O is called the pole of the line with respect to the conic.

99. If a point B is on the polar of O, then it is harmonically conjugate to O with respect to the two intersections K and M of the line BC with the conic.  But for the same reason O is on the polar of B.  We have, then, the fundamental theorem

If one point lies on the polar of a second, then the second lies on the polar of the first.

100.  Conjugate points and lines. Such a pair of points are said to be conjugate with respect to the conic.  Similarly, lines are said to be conjugate to each other with respect to the conic if one, and consequently each, passes through the pole of the other.

[Figure 27]

FIG. 27

101.  Construction of the polar line of a given point. Given a point P, if it is within the conic (that is, if no tangents may be drawn from P to the conic), we may construct its polar line by drawing through it any two chords and joining the two points of intersection of the two pairs of tangents at their extremities.  If the point P is outside the conic, we may draw the two tangents and construct the chord of contact (Fig. 27).