It follows that v w 0 and p 0 0w v ww in this case

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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 O=F 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 r = the distance from P to L d + r cos(θ) so that r = e(d + r cos(θ)). Solving this equation for r, yields ed r= 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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