137. The intersections of corresponding rays of a pencil of the second order in involution are all on a straight line u, and the intersection of any two tangents ab, when joined to the intersection of the corresponding tangents a'b', gives a line which passes through a fixed point U, the pole of the line u with respect to the conic.
138. Involution of rays determined by a conic. We have seen in the theory of poles and polars (§ 103) that if a point P moves along a line m, then the polar of P revolves about a point. This pencil cuts out on m another point-row P', projective also to P. Since the polar of P passes through P', the polar of P' also passes through P, so that the correspondence between P and P' is double. The two point-rows are therefore in involution, and the double points, if any exist, are the points where the line m meets the conic. A similar involution of rays may be found at any point in the plane, corresponding rays passing each through the pole of the other. We have called such points and rays conjugate with respect to the conic (§ 100). We may then state the following important theorem:
139. A conic determines on every line in its plane an involution of points, corresponding points in the involution [pg 82] being conjugate with respect to the conic. The double points, if any exist, are the points where the line meets the conic.
140. The dual theorem reads: A conic determines at every point in the plane an involution of rays, corresponding rays being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic.
PROBLEMS
1. Two lines are drawn through a point on a conic so as always to make right angles with each other. Show that the lines joining the points where they meet the conic again all pass through a fixed point.