Stitz-Zeager_College_Algebra_e-book

This result is generalized in the following theorem

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Unformatted text preview: ly, we verify that one focus is at (0, 0), and the formula given in Exercise 7 in Section 7.5 gives the eccentricity is e in this case as well. If e = 1, the equation r = 1−eed θ) reduces to cos( r= d 1−cos(θ) which gives the rectangular equation y 2 = 2d x + d 2 . This is a parabola with vertex −d, 0 2 opening to the right. In the language of Section 7.3, 4p = 2d so p = d , the focus is (0, 0), 2 the focal diameter is 2d and the directrix is x = −d, as required. Hence, we have shown that in all cases, our ‘new’ understanding of ‘conic section’, ‘focus’, ‘eccentricity’ and ‘directrix’ as presented in Definition 11.1 correspond with the ‘old’ definitions given in Chapter 7. Before we summarize our findings, we note that in order to arrive at our general equation of a conic r = 1−eed θ) , we assumed that the directrix was the line x = −d for d > 0. We could have just as cos( easily chosen the directrix to be x = d, y = −d or y = d. As the reader can verify, in these cases we obtain the forms r = 1+eed θ) ,...
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