PP 11.5 - Conic Sections Conic sections, or conics, get...

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Conic Sections
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Conic sections, or conics, get their name because they result from intersecting a cone with a plane.
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PARABOLA A parabola is the set of points in a plane that are equidistant from a fixed point F (the focus) and a fixed line (the directrix). The vertex is the point halfway between the focus and the directrix.
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The line through the focus perpendicular to the directrix is called the axis of symmetry of the parabola.
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Equation of a Parabola To obtain an equation for a parabola, we place its vertex at the origin O and its directrix parallel to the x -axis. We will use the definition of parabola to get its equation.
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The defining property of a parabola is that the distance to the focus equals the distance to the directrix. 2 2 ( ) x y p y p + - = +
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Hence, 2 2 2 2 2 2 2 2 2 2 ( ) ( ) 2 2 4 x y p y p y p x y py p y py p x py + - = + = + + - + = + + = 2 2 ( ) x y p y p + - = +
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Similar arguments yield the equations of the other three possibilities. Vertical Parabolas
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Horizontal Parabolas; notice that these curves do NOT represent functions.
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Example Find the focus and directrix of the parabola y 2 + 10 x = 0 and sketch the graph. 2 5 2 5 2 ) ( 4 10 - = - = - = p x x y 2 5 2 5 : ), 0 , ( = - = x D F Horizontal parabola
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ELLIPSE An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. These points are called the foci. Q
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To obtain the equation for an ellipse, we place the foci on the x -axis at the points (– c , 0) and ( c , 0) so that the origin is halfway between the foci.
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P ( x , y ) is a point on the ellipse, the definition says that |PF 1 | + | PF 2 | = 2 a Let the sum of the distances from a point on the ellipse to the foci be 2 a > 0. 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 11.5 - Conic Sections Conic sections, or conics, get...

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