Unformatted text preview: ess outlined in Section 11.4 gives6
1 − e2
e2 d2 2 x− e2 d
1 − e2 2 + 1 − e2
e2 d2 y2 = 1
e2 d
,0
1−e2
2 d2
e
and
(1−e2 )2 We leave it to the reader to show if 0 < e < 1, this is the equation of an ellipse centered at
with major axis along the xaxis. Using the notation from Section 7.4, we have a2 =
22 e
b2 = 1−d 2 , so the major axis has length 12ed2 and the minor axis has length √2ede2 . Moreover, we ﬁnd
e
−e
1−
that one focus is (0, 0) and working through the formula given in Deﬁnition 7.5 gives the eccentricity
2d
to be e, as required. If e > 1, then the equation generates a hyperbola with center 1e e2 , 0 whose
−
2 2 h
transverse axis lies along the xaxis. Since such hyperbolas have the form (x−2 ) − y2 = 1, we need
a
b
2 d2
2 d2
to take the opposite reciprocal of the coeﬃcient of y 2 to ﬁnd b2 . We get7 a2 = (1e e2 )2 = (ee −1)2 and
2
−
22 22 e
ed
b2 = − 1−d 2 = e2 −1 , so the transverse axis has length e2ed1 and the conjugate axis has length √2ed 1 .
2−
e
e2 −
Additional...
View
Full Document
 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

Click to edit the document details