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Engineering Calculus Notes 135

Engineering Calculus Notes 135 - the line is the directrix...

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2.1. CONIC SECTIONS 123 (the diameter ) and derive as the equation of the ellipse y 2 = | VS | x = p parenleftBig 1 x d parenrightBig x = px p d x 2 . (2.6) Hyperbolas: In the final case, when π 2 α<φ π 2 , The same arguments as in the ellipse case yield Equation ( 2.5 ), but this time the segment VS exceeds PL ; the Greek for “excess” is ἠπερβολή , and γ is called a hyperbola . A verbatim repetition of the calculation leading to Equation ( 2.6 ) leads to its hyperbolic analogue, y 2 = px + p d x 2 . (2.7) P lies between V and P . The Focus-Directrix Property Pappus, in a section of the Collection headed “Lemmas to the Surface Loci 9 of Euclid”, proves the following ([ 25 , p. 153]): Lemma 2.1.1. If the distance of a point from a fixed point be in a given ratio to its distance from a fixed straight line, the locus of the point is a conic section, which is an ellipse, a parabola, or a hyperbola according as the ratio is less than, equal to, or greater than, unity. The fixed point is called the
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Unformatted text preview: , the line is the directrix , and the ratio is called the eccentricity of the conic section. This focus-directrix property of conics is not mentioned by Apollonius, but Heath deduces from the way it is treated by Pappus that this lemma must have been stated without proof, and regarded as well-known, by Euclid. We outline a proof in Appendix B . The focus-directrix characterization of conic sections can be turned into an equation. This approach—treating a curve as the locus of an equation in the rectangular coordinates—was introduced in the early seventeenth century by Ren´ e Descartes (1596-1650) and Pierre de ±ermat (1601-1665). Here, we sketch how this characterization leads to equations for the conic sections. 9 This work, like Euclid’s Conics , is lost, and little information about its contents can be gleaned from Pappus....
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