Conic Sections

Definition of the Conic Sections

Let e > 0 (called the eccentricity) and let F be a point (called the Focus) in the plane and let l be a fixed line (called the directrix).  Then a conic is the set of points, P, such that

                 = e


Where PF is the distance from the point to the focus and d(Pl) is the distance from the point to the line..  We have the following classification:

   e Type
0 circle
0< e <1 ellipse
1 parabola
e > 1 hyperbola


The Circumference of an Ellipse


Find the circumference of the ellipse

            x2          y2  
                    +         =  1
            4           9

Solution:  We can solve for y


Soon we will see it is a square that we want so that we can ignore the sign.


So that


Now the find the length, we integrate this equation from 0 to 2 and then multiply by 4.


To attempt to solve this integral we let

        x  =  2 sin q,       dx  =  2 cos q

The integral reduces to


This integral is anything but elementary, but we can approximate it with a computer as


and doubling it gives



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