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

302

ANALYTIC GEOMETRY

The student should note the symmetry of formulas (6) and
(7) in (x1, y1) and (xa', y1'). The results embodied in these
formulas we state as a theorem.
THEOREM 9. If (x1, yJ) is one extremity of a diameter D
meeting the hyperbola (5 a), and (x1', yl') is a properly chosen one
of the extremities of the conjugate diameter D', then (6) are the
equations of D and D', and (7) give the relations between the two
extremities.
Let 2 al be the length of D and 2 bl that of D'. It can be
shown that
8 a12 = e2X12 - b2 = e2x1t2 + a2,
(8) l b12 = e2x12 — a2 = e2x/2 + b2.
Hence a2 - b12 = a2 - b2
and we have the following theorem.
THEOREM 10. The difference of the squares of the lengths of
any two conjugate diameters of a hyperbola is constant, and equals
the difference of the squares of the axes.
EXERCISES
Establish the following formulas.
1. Formulas (3). 3. Formulas (7).
2. Formulas (4). 4. Formulas (8).
5. Show that two conjugate diameters of a hyperbola are
never equal unless the hyperbola is rectangular and that in
this case they are always equal.
6. Prove that the product of the focal radii to any point of
an ellipse equals the square of half the diameter conjugate to
the diameter through the point.
7. State and prove the corresponding theorem for the
hyperbola.
7. Physical Meaning of Conjugate Diameters. Ellipse.
Consider a flat bar of iron, on the end ABCD of which a
circle is drawn. Let D and D' be any two mutually perpen