This preview shows page 1. Sign up to view the full content.
Unformatted text preview: n pictures we have, Hooked on Conics Minor Axis 420 Major Axis V1 F1 F2 C V2 An ellipse with center C ; foci F1 , F2 ; and vertices V1 , V2
Note that the major axis is the longer of the two axes through the center, and likewise, the minor
axis is the shorter of the two. In order to derive the standard equation of an ellipse, we assume that
the ellipse has its center at (0, 0), its major axis along the x-axis, and has foci (c, 0) and (−c, 0)
and vertices (−a, 0) and (a, 0). We will label the y -intercepts of the ellipse as (0, b) and (0, −b) (We
assume a, b, and c are all positive numbers.) Schematically,
y (0, b) (x, y ) x (−a, 0) (−c, 0) (c, 0) (a, 0) (0, −b)
Note that since (a, 0) is on the ellipse, it must satisfy the conditions of Deﬁnition 7.4. That is, the
distance from (−c, 0) to (a, 0) plus the distance from (c, 0) to (a, 0) must equal the ﬁxed distance
d. Since all of these points lie on the x-axis, we get
distance from (−c, 0) to (a, 0) + distance from (c, 0) to (a, 0) = d
(a + c) + (a − c) = d
2a = d 7.4 Ellipses 421 In other words, the ﬁxed distance d mentioned in the deﬁnition of the ellipse is none other than
the length of the major axis. We now use...
View Full Document