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Unformatted text preview: 126 CHAPTER 2. CURVES Ellipses
When e = 1, completing the square in Equation (2.8) and moving the
y -axis to the right k/(1 − e2 ) units, we obtain
(1 − e2 )x2 + y 2 = k2 e2
1 − e2 (2.11) as the equation of the conic section with eccentricity e = 1, directrix where
1 − e2 and focus
, 0) = F (
e2 − 1
1 − e2 Noting that the x-coordinate of the focus is e2 times the constant in the
equation of the directrix, let us consider the case when the focus is at
F (−ae, 0) and the directrix is x = −a/e: that is, let us set
1 − e2 This is positive provided 0 < e < 1, the case of the ellipse.
If we divide both sides of Equation (2.11) by its right-hand side, we
recognize the resulting coeﬃcient for x2 as 1/a2 and get the equation
a (1 − e2 ) (2.12) In the case e < 1, we can deﬁne a new constant b by
b=a 1 − e2 , so that (2.12) becomes
x2 y 2
+ 2 = 1.
This is the equation of the ellipse with focus F (−ae, 0) and directrix
x = −a/e, where
e= 1− b
a 2 . Let us brieﬂy note a few features of this curve: (2.14) ...
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