Verification: The reciprocal of the length of two focal radii on a focus chord on an ellipse is a constant value.

let f be the focus, l the corresponding directrix, and AB the focus chord. AP,Q,R BQ and FR are perpendicular to L, and their vertical feet are P, Q and R..

defined by conic curve, AF = e * AP, BF = e * BQ.

in trapezoidal ABQP, the ratio AF/BF is known, It can be found that:

fr = af/ab * bq+BF/ab * AP

= af/(af+BF)/e * BF/(af+BF)/e * af

= 2af * BF/(af+BF)/e < p. BF

Method 2: Use the polar coordinate formula: r = ep/(1-e*cosθ).

The focal radii R1 and R2 correspond to θ, θ+pi

so 1/R1+1/R2 = (1-e * cos θ)/EP+(1+e * cos θ.