Connecting Polar to Rectangular Consider the hyperbola
Chapter 8, Problem 8.1.1.406(choose chapter or problem)
Connecting Polar to Rectangular Consider the hyperbola
\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\),
where half the length of the transverse 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 hyperbola. [Hint: Prove that PF/PD is the constant c/a, where P is a point on the hyperbola, 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 ea - a/e.
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