Connecting Polar to Rectangular Consider the ellipse where

Chapter 8, Problem 8.1.1.405

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Connecting Polar to Rectangular Consider the ellipse

              \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\),

where half the length of the major axis is a, and the foci are \((\pm c,\ 0)\) such that \(c^{2}=a^{2}-b^{2}\). Let L be the vertical line \(x=a^{2} / c\).

(a) Prove that L is a directrix for the ellipse. [Hint: Prove that PF/PD is the constant c/a, where P is a point on the ellipse, and D is the point on L such that PD is perpendicular to L.]

(b) Prove that the eccentricity is e = c/a.

(c) Prove that the distance from F to L is a/e - ea.

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