Engineering Calculus Notes 140

Engineering Calculus Notes 140 - y-axis. In particular,...

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128 CHAPTER 2. CURVES b F F x = a/e x = a/e a b c a Figure 2.7: The ellipse x 2 a 2 + y 2 b 2 = 1, a > b > 0 Hyperbolas: When e > 1, then the coefcient o± y 2 in Equation ( 2.12 ) is negative ; in this case we de²ne the number b > 0 by b = a r e 2 1 and obtain the hyperbolic analogue o± Equation ( 2.13 ) x 2 a 2 y 2 b 2 = 1 (2.15) as the equation o± the hyperbola with ±ocus F ( ae, 0) and directrix x = a/e , where e = R 1 + p b a P 2 . This curve shares some ±eatures with the ellipse, but is dramatically di³erent in other respects: As in Equation ( 2.13 ), x and y each enter Equation ( 2.15 ) only as their squares, so replacing x with x (or y with y ) does not change the equation: this means the curve is invariant under re´ection across the
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Unformatted text preview: y-axis. In particular, this gives us a second ocus/directrix pair or the curve: F ( ae, 0) and x = a/e . Equation ( 2.15 ) orces | x | a : thus there are no y-intercepts: the curve has two separate branches, one opening to the right rom ( a, 0) and the other opening to the let rom ( a, 0); these are the vertices o the hyperbola. The distance 2 a between the vertices is called the transverse axis o the hyperbola. As | x | grows, so does | y | : in each branch, y ranges over all values, with x decreasing to x = a in the...
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