PP 11.6 - Slide 11 Conic Sections in Polar Coordinates...

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Conic Sections in Polar Coordinates Slide 11
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Theorem Let F be a fixed point (called the focus ) and let L be a fixed line (called the directrix) . Let e be a fixed positive number (called the eccentricity ). The set of all points P in the plane such that e PL PF = | | | | is a conic section. The conic is (a) an ellipse if e < 1 (b) a parabola if e = 1 (c) a hyperbola if e > 1.
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Proof of Theorem If e = 1, we get |PF| = |PL|, which is the definition of parabola, so that case is proven. Consider the situation depicted on the left. F = (0, 0), L: x = d, P = (r, θ) in polar coordinates.
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|PF| = r |PL| = d – rcos(θ) e PL PF = | | | | OR r = e[d – rcos(θ)] From the graph: From the theorem:
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r = e[d – rcos(θ)] ) ( 2 2 x d e y x - = + ) 2 ( 2 2 2 2 2 x dx d e y x + - = + 2 2 2 2 2 2 2 ) 1 ( d e y x de x e = + + - Completing the square on the left: 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 2 - + - = - + -
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

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PP 11.6 - Slide 11 Conic Sections in Polar Coordinates...

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