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Engineering Calculus Notes 133

# Engineering Calculus Notes 133 - applied to PL with width...

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2.1. CONIC SECTIONS 121 B C Q R V P Figure 2.3: Conic Section There are three possibilities: Parabolas: If PV is parallel to AC , then P is the only vertex of γ . We wish to relate the square of the ordinate, | QV | 2 = | QV |·| VR | , to the abcissa | PV | . By Equation ( 2.2 ), the square of the ordinate equals | BV |·| VC | . Apollonius constructs a line segment PL perpendicular to the abcissa PV , called the orthia 7 : he then formulates the relation between the square of the ordinate and the abcissa 8 as equality of area between the rectangle LPV and a square with side | QV | (recall Equation ( 2.2 )). | QV | 2 = | PL || PV | . (2.3) In a terminology going back to the Pythagoreans, this says that the square on the ordinate is equal to the rectangle applied
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Unformatted text preview: applied to PL , with width equal to the abcissa. Accordingly, Apollonius calls this curve a parabola (the Greek word for “application” is παραβολή ) [ 25 , p. 359]. If we take rectangular coordinates in P with the origin at P and axes parallel to QV ( y = | QV | ) and PV ( x = | PV | ), then denoting the length of the orthia PL by p , we obtain the equation for the rectangular coordinates of Q y 2 = px. (2.4) 7 the Latin translation of this term is latus rectum , although this term has come to mean a slightly diFerent quantity, the parameter of ordinates. 8 Details of a proof are in Appendix A...
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