Calculus: Analytic Geometry and Calculus, with Vectors

6.2 Geometry of cones and conics 365

Putting this value of xl in (6.261) gives the equation

(6.271) (1 - e2)x2 + y2 =

e2p2

-e2

This equation of the conic is a good source of information, but we obtain

a better sourceby putting the equation in the "standard form." To do

this neatly and correctly, we sacrifice some paper to put (6.271) in the

form

(1 - e2)x2 y2 e2p2

1 +^1 1-7

divide the numerator and denominator of the first term by (1 - e2) to

obtain

x2 y2 e2p2

(^1) + 1 1-e
1 - e2
and then divide both members of this equation by the right member to
obtain an equation which is put in the form
2
(6.272)
x
e2p2 +
y
e2p2 =1, (ellipse)
(1 - e2)2 1 - e2
when 0 < e < 1, and in the form
(6.273) e2p2 - e p2 = 1, (hyperbola)
(e2 - 1)2 e2 - 1
when e > 1. Everything is so arranged that the denominators in (6.272)
and (6.273) are positive.
Unless we think a bit about sections of cones, we cannot fully appreci-
ate the significance of these formulas. It is sometimes said that any
reasonably sane person should feel quite sure that an ellipse is an egg-
shaped oval which has a "small end" at the part of the ellipse nearest the
vertex of the cone and which has a "big end" at the part of the ellipse
farthest from the vertex of the cone. However, (6.272) shows very
clearly that the x and y axes are axes of symmetry of the ellipse and that
the origin is a center (center of symmetry) of the ellipse. Thus (6.272)
reveals the astonishing fact that the ellipse has a center and that the two
"ends" of the ellipse are alike (or congruent). Similarly, suppose a
particular hyperbola H intersects one nappe of a cone at some points
near the vertex of the cone but intersects the other nappe only at points
very far from the vertex of the cone. It would seem to be incredible that
the two branches of this hyperbola should be alike, but they are alike.