WOLFRAM|DEMONSTRATIONS PROJECT

An Elliptic Identity

semi-major axis a

1.7

semi-minor axis b

1.1

EF × EG ×

2

OP

=3.5

2

a

2

b

=3.5

Let E be a point on an ellipse, O its center, F and G its two foci, and P the point on the tangent line to E such that OP is perpendicular to the tangent line. Then the identity is

EF×EG

×

2

OP

=

2

a

2

b

. Drag the point E to see that the identity holds for ellipses of different shapes and sizes.