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

290

ANALYTIC GEOMETRY

EXERCISES
1. Find the locus of the mid-points of the chords of the
ellipse
3 + 4y2 = 12,
which are inclined at an angle of 135~ to the axis of x. First
draw an accurate figure, showing the chords and the locus; then
solve the problem analytically, using the method, but not the
formulas, of the text.
2. Prove the converse of Theorem 1, namely, that every
diameter of the ellipse (1) bisects some set of parallel chords.
Show that, if X'(* 0) is the slope of the diameter, then the
chords which it bisects are of slope X, where
b2
a2X'
3. Prove analytically that the tangent to an ellipse at an
extremity of a diameter is parallel to the chords which the
diameter bisects.
Suggestion. Let (x1, yl) be the coordinates of the extremity
of the diameter and find, by using (6), the slope X of the
chords in terms of x, and yi.
2. Conjugate Diameters of an Ellipse. Two mutually perpendicular diameters of a circle have the property that each
bisects the chords parallel to the other. The axes of an ellipse have this same property. Are there other pairs of
diameters of the ellipse which have it? This question is
answered in the affirmative by the following theorem.
TTHEOREM 2. If one diameter bisects the
chords parallel to a second, the second diameter bisects the chords parallel to the first.
The two diameters stand in a reciprocal
FIG 2 relationship; each bisects the chords parallel to the other. We call them a pair of
conjugate diameters, and say that each is conjugate to the
other.