PP Section 11.4 - The Ellipse Definition An ellipse is the...

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The Ellipse
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Definition 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.
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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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If 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 2 2 2 ( ) ( ) 2 x c y x c y a + + + - + =
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2 2 2 2 ( ) ( ) 2 x c y x c y a + + + - + = 2 2 2 2 ( ) 2 ( ) x c y a x c y - + = - + + To simplify algebraic operations, let’s place a radical on each side of the equation. Squaring both sides, we have: 2 2 2 2 2 2 2 2 2 2 4 4 ( ) 2 x cx c y a a x c y x cx c y - + + = - + + + + + +
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2 2 2 ( ) a x c y a cx + + = + After simplifying, the last expression becomes: 2 2 2 2 4 2 2 2 ( 2 ) 2 a x cx c y a a cx c x + + + = + + Squaring both sides again to eliminate the radical: After simplifying, the last expression becomes: 2 2 2 2 2 2 2 2 ( ) ( ) a c x a y a a c - + = -
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In Geometry we learned that in any triangle the sum of the lengths of two sides is greater that the length of the third side. Consider the triangle determined by the foci and P.
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