Engineering Calculus Notes 141

Engineering Calculus Notes 141 - y/x = b/a . It is...

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2.1. CONIC SECTIONS 129 right branch (increasing to x = a in the left branch) as y decreases from to zero and then increasing ( resp . decreasing) again as y passes zero and goes to −∞ . This time, there is no a priori inequality between a and b . Replacing the number 1 on the right of Equation ( 2.15 ) with 1 x 2 a 2 y 2 b 2 = 1; (2.16) amounts to interchanging the roles of x ( resp . a ) and y ( resp . b ): y 2 b 2 x 2 a 2 = 1 (2.17) The locus of this equation is a curve whose branches open up and down from (0 ,b ) and (0 , b ) respectively; their foci are at (0 ,be ) ( resp . (0 , be )) and their directrices are y = b/e ( resp . y = b/e ), where the eccentricity is e = r 1 + p a b P 2 . The equation obtained by replacing the number 1 with 0 on the right of Equation ( 2.15 ) x 2 a 2 y 2 b 2 = 0 (2.18) has as its locus the two lines
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Unformatted text preview: y/x = b/a . It is straightforward to check that as a point P ( x,y ) moves along the hyperbola with x , the ratio y/x tends to b/a , so the distance of P from one of these lines goes to zero. These lines are called the asymptotes of the hyperbola. The distance from the origin to the two foci is given by the formula c = | ae | = R a 2 + b 2 A consequence of this formula is another characterization of a a hyperbola (Exercise 9 ): The (absolute value of the) diFerence between the distances of any point on the hyperbola to the two foci equals the transverse axis. This information is illustrated in Figure 2.8 ....
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