Calculus: Analytic Geometry and Calculus, with Vectors

6.3 Ellipses 369

intersects the conic at two points, then the mid-point of the chord ofthe conic
joining these points has coordinates (x,y), where

_ m 2+2(A-m2)

x-b2(A-m2)' y -b

IM'

2(!1 -m2)

Use this result to show that, for each fixed m, the set D of mid-points of chords

having slope m lies on a line through the center of the conic. Remark: This set

D, which is always a line segment when the conic is a circle or an ellipse and is

sometimes a whole line when the conic is a hyperbola, is called a diameter of the

conic.

10 Let m be a positive constant. A surface S contains the origin, and when

y 6 0, it contains the circlewhich lies in a plane parallel to the xz plane and has

a diameter coincidingwith the line segment joining the two points (O,y,O) and

(0,y,2my) in the yz plane. Sketch a figure showing S and show that the equation

of S is

x2 + (z - -y)2 = m2y2.

Remark: One who wishes to rise above minimum requirements may show that

(i) S is a quadric surface, (ii) S is a cone and hence is a quadric cone, (iii) S is

not a right circular cone. One who wishes to rise to still greater heights may

try to decide whether we know enough to determine whether the cone has an

axis and, if so, whether sections made by planes perpendicular to this axis are

ellipses.

6.3 Ellipses Remarkable geometric properties of ellipses can be

extracted from Figure 6.31. This

figure, like Figure 6.22, shows a

cone having a vertical axis. The

axis lies in the plane of the paper.

The plane 7r intersects the cone in

an ellipse E of which the two points

(vertices, in fact) Yl and Y2 lie in

the plane of the paper. The smaller

circle represents a sphere which is

tangent to the cone at the points of

a circle which determines the plane

7r1 and is tangent to a atFl. As we

saw in Section 6.2, al and ir inter-

sect in a line LI and, moreover, Fl is

a focus and Ll is a directrix of the

ellipse E. The larger circle repre-

sents a sphere which is tangent to

the cone at the points of a circle

which determines the plane 72 and

Figure 6.31

is tangent to it at F2. The planes 7r2 and ,r intersect in a line L2, and the

same procedure which was applied to F, and L, shows that F2 is another