For a point on an ellipse, let be the distance from the center of the ellipse to the

Chapter 10, Problem 97

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For a point P on an ellipse, let  \(d\)  be the distance from the center of the ellipse to the line tangent to the ellipse at P. Prove that  \(\left(P F_{1}\right)\left(P F_{2}\right) d^{2}\)  is constant as P varies on the ellipse, where  \(P F_{1}\)  and  \(P F_{2}\)  are the distances from P to the foci  \(F_{1}\)  and  \(F_{2}\)  of the ellipse.

Text Transcription:

d

(PF_1)(PF_2) d^2

PF_1

PF_2

F_1

F_2