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

DIAMETERS. POLES AND POLARS

299

5. Diameters of a Parabola. When one focus and the corresponding directrix of a central conic —an ellipse or a hyperbola-are held fast and the center is allowed to recede indefinitely along the transverse axis, the limit of the conic is a
parabola and the limit of the diameters of the conic is a set
of lines parallel to the axis of the parabola.
Accordingly, by a diameter of a parabola we
shall mean any line in the direction of the axis /
of the parabola.
If this definition is really in accord with/ x
that of a diameter of a central conic, we
should find that the mid-points of a set of
parallel chords of a parabola lie on a line in
the direction of the axis. This is the case. FIG. 8
If the chords are perpendicular to the axis,
their mid-points evidently lie on the axis; if the slope of the
chords is X(=/ 0), and the equation of the parabola is
(1) y2 = 2mx,
the mid-points of the chords lie on the line
(2) Y=
as may easily be shown.
EXERCISES
1. Establish the result embodied in formula (2).
2. What is the equation of the diameter of the parabola
y2 + 6x =0,
which bisects the chords of slope ~?
3. Prove Ex. 3, ~ 1, for the parabola.
4. There are no conjugate diameters for a parabola. Why?
6. Extremities and Lengths of Conjugate Diameters. Ellipse.
Let the coordinates of one extremity of a diameter D (not an
axis) of the ellipse
+2 -
~~~(1)~~a + b2