Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

322

ANALYTIC GEOMETRY

between the asymptotes not containing a branch of the hyper.
bola. In this case, too, the theorem is valid, but the points
P1 and P2, instead of being on the same side of 0, are on
opposite sides.*
COROLLARY. The points P1 and P2 are on the same or opposite sides of 0, according as D meets or does not meet the conic.
Given, now, a point, P, and its polar, L, with respect to a
central conic. If P traces a diameter D, then L moves always
parallel to the conjugate diameter.
In the case that D intersects the conic, and P is an intersection, L is the tangent at P. If P then moves in along D
toward the center as its limit, L ceases to meet the conic, and
recedes indefinitely. On the other hand, if P moves out
along D, receding indefinitely, L moves in toward the center,
and approaches the diameter conjugate to D as its limit.
The case in which the conic is a hyperbola and D intersects
the conjugate hyperbola remains. If P is at one of the intersections, L is the tangent to the conjugate hyperbola at the
other; cf. Ex. 6. If P then moves in toward the center, L
moves away from it, and so forth, as before.
Parabola. Corresponding to Theorem 9, we have the fol2 lowing theorem, the proof of which is left to
the student.
/P/ - D THEOREM 10. Let P1 and L1 be pole and
polar in a parabola, and let the diameter
through P1 meet L1 in P2 and the parabola in
Q. Then L1 is parallel to the tangent at Q,
and Q is the mid-point of PP?2.
FIG. 28 Consequently, if a point P traces a diameter
* Let the student give an analytical proof of these facts and hence of
the corollary; cf. Exs. 1, 2. There is no geometrical proof, analogous
to that of the text. The r6les of Q1 and Q2 in that proof cannot be
played here by the points in which D meets the conjugate hyperbola;
these points do not divide P1P2 harmonically: L1 is the polar of P1
with respect to the given hyperbola, and not with respect to its conjugate.