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

DIAMETERS. POLES AND POLARS

295

First draw an accurate figure, showing the chords and the
diameter; then solve the problem analytically, without reference to the formulas of the text.
2. If a set of parallel chords has a slope nearly equal to
that of an asymptote S, then the diameter D bisecting the
chords has a slope nearly equal to that of S, and when the
chords approach a limiting position of parallelism to S, then
D approaches S as its limit. Draw a figure showing the reasonableness of this theorem and then prove the theorem analytically by use of (3).
3. Prove the converse of Theorem 4, namely: Every
diameter of a hyperbola, not an asymptote, bisects some set
of parallel chords. Cf. ~ 1, Ex. 2.
4. Show that the mid-point of a chord of a hyperbola is also
the mid-point of the chord of the conjugate hyperbola which
lies on the same line. Hence show that the mid-points of the
chords of a given slope lie on one and the same diameter,
whether the chords are chords of the given hyperbola or of
its conjugate.
5. Prove Ex. 3, ~ 1, for a hyperbola.
4. Conjugate Diameters of a Hyperbola. Let the diameter,
D', of slope X', bisect the chords of the hyperbola
(1) - -1
a2 b5t
which are parallel to the diameter D, of slope X(# 0). Then,
by Th. 4, ~ 3,
b2 b?
AX' _ or XX-' =-.
a2X a2
Since these equations are symmetric in A and A', it follows that
the diameter D bisects the chords parallel to D'.
Thus Theorem 2, ~ 2, is established for the hyperbola, and
the two diameters D and D' are, in the sense of that theorem,
conjugate diameters; each bisects the chords parallel to the
other.