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Let C be the ellipse with equation x/a+y/b=1, with a>b, and let F,F'=(±,0) be its foci (see Figure 7.1.2 ).
(a cos , b sin ).
The area of the shaded sector below is
The length of the arc from (a,0) to the point (a cos , b sin ) is given by a( E(pi/2,e) - E(pi/2 - theta, e) ), where E is an elliptic integral (see the Standard Math Tables and Formulas for elliptic integrals). Setting =2 we get
area C= ab,
perimeter C=4a E(pi/2,e).
With respect to a coordinate system with origin at a focus, the equation is
where l=b/a is half the latus rectum. (Use the + sign for the focus with positive x-coordinate and the - sign for the other.)
Wed Oct 4 16:41:25 PDT 1995
This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.