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

 PF                  = e      d(Pl)

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

Example

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

15.87

and doubling it gives

31.74