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Determine the eccentricity and distance from the focus to the directrix of a conic section

by RoRi

Given the following polar equation

    \[ r = \frac{2}{1+ \cos \theta} \]

for a conic section with a focus F at the origin and vertical directrix lying to the right of F, determine the eccentricity e and the distance d from the focus to the directrix.

By Theorem 13.18 we know the points on a conic section satisfy the polar equation

    \[ r = \frac{ed}{1 + e \cos \theta}. \]

Therefore, e = 1 and d = 2.

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