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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
**parametric representation**for*C*is given by(

*a*cos ,*b*sin ).The

**area**of the shaded sector below is½

*ab*=½*ab*arccos(*x*/*a*).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 getarea

*C*=*ab*,

perimeter*C*=4*a**E*(pi/2,*e*). - A
**rational parametric representation**for*C*is given by - The
**polar equation**for*C*in the usual polar coordinate system isWith 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.) - Let
*P*be any point of*C*. The**sum of the distances***PF*and*PF'*is constant, and equal to 2*a*. - Let
*P*be any point of*C*. Then the rays*PF*and*PF'*make the same angle with the tangent to*C*at*P*. Thus any light ray originating at*F*and reflected in the ellipse will also go through*F'*. - Let
*T*be any line tangent to*C*. The product of the distances from*F*and*F'*to*T*is constant, and equals*b*. **Lahire's theorem**: Let*D*and*D'*be fixed lines in the plane, and consider a third, moving, line on which three points*P*,*P'*and*P''*are marked. If we constrain*P*to lie in*D*and*P'*to lie in*D'*, then*P''*describes an ellipse.

*Silvio Levy
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.