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Unformatted text preview: ances involved were
measurements from the center to a focus and from the center to a vertex. One way to reconcile the
‘old’ ideas of focus, directrix and eccentricity with the ‘new’ ones presented in Deﬁnition 11.1 is
to derive equations for the conic sections using Deﬁnition 11.1 and compare these parameters with
what we know from Chapter 7. We begin by assuming the conic section has eccentricity e, a focus
F at the origin and that the directrix is the vertical line x = −d as in the ﬁgure below.
y d r cos(θ)
P (r, θ) r θ
x = −d x 11.6 Hooked on Conics Again 835 Using a polar coordinate representation P (r, θ) for a point on the conic with r > 0, we get
e= the distance from P to F
the distance from P to L
d + r cos(θ) so that r = e(d + r cos(θ)). Solving this equation for r, yields
1 − e cos(θ)
At this point, we convert the equation r = e(d + r cos(θ)) back into a rectangular equation in the
variables x and y . If e > 0, but e = 1, the usual conversion proc...
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